%!PS-Adobe-2.0 %%Creator: dvips(k) 2020.1 Copyright 2020 Radical Eye Software %%Title: arXiv:2006.01810v3 [math.AG] 14 May 2022 %%CreationDate: Tue May 17 02:39:27 2022 %%Pages: 37 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: EUFM10 %%DocumentPaperSizes: a4 %%EndComments %%BeginDefaults %%ViewingOrientation: 1 0 0 1 %%EndDefaults %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z SL4-torus-knots-45.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2022.05.17:0239 %%BeginProcSet: texc.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix 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FK(Let)21 b(\000)f(b)s(e)g(a)h(\014nitely)d(generated)j(group)f(and)g (let)g FD(G)h FK(b)s(e)f(an)g(algebraic)g(group)g(o)m(v)m(er)h(an)g (algebraically)125 2252 y(closed)38 b(\014eld)f FD(k)k FK(of)d(zero)h(c)m(haracteristic.)66 b(The)37 b(space)i FD(R)q FK(\(\000)p FD(;)15 b(G)p FK(\))40 b(of)e(represen)m(tations)g FD(\032)h FK(:)f(\000)h FC(!)f FD(G)125 2360 y FK(forms)29 b(an)i(algebraic)f(v)-5 b(ariet)m(y)31 b(kno)m(wn)f(as)g(the)h (represen)m(tation)f(v)-5 b(ariet)m(y)d(.)42 b(Moreo)m(v)m(er,)33 b(if)c(w)m(e)i(w)m(an)m(t)h(to)125 2467 y(parametrize)h(isomorphism)d (classes)j(of)h(represen)m(tations,)g(w)m(e)f(need)g(to)h(consider)e (the)h(Geometric)125 2575 y(In)m(v)-5 b(arian)m(t)30 b(Theory)g(\(GIT\))g(quotien)m(t)1387 2726 y FB(M)p FK(\(\000)p FD(;)15 b(G)p FK(\))27 b(=)e FD(R)q FK(\(\000)p FD(;)15 b(G)p FK(\))21 b FA(\014)f FD(G;)125 2877 y FK(where)33 b FD(G)i FK(acts)g(on)f FD(R)q FK(\(\000)p FD(;)15 b(G)p FK(\))35 b(b)m(y)f(conjugation.)53 b(This)32 b(giv)m(es)j(rise)e(to)i (the)f(mo)s(duli)e(space)i(of)h(repre-)125 2985 y(sen)m(tations)30 b(of)h(\000)f(in)m(to)g FD(G)p FK(,)h(also)g(called)e(the)i FD(G)p FK(-c)m(haracter)i(v)-5 b(ariet)m(y)30 b(of)h(\000.)224 3128 y(Character)41 b(v)-5 b(arieties)40 b(are)h(v)m(ery)g(ric)m(h)e (ob)5 b(jects)41 b(that)h(con)m(tain)e(subtle)g(geometric)h (information)125 3236 y(linking)25 b(distan)m(t)i(areas)i(in)d (mathematics.)41 b(An)27 b(imp)s(ortan)m(t)g(instance)h(is)f(when)g(w)m (e)h(tak)m(e)i(\000)25 b(=)g FD(\031)3436 3250 y FJ(1)3475 3236 y FK(\(\006\))125 3344 y(to)32 b(b)s(e)e(the)i(fundamen)m(tal)e (group)h(of)g(the)h(compact)g(orien)m(table)f(surface)g(of)h(gen)m(us)f FD(g)s FK(.)44 b(In)31 b(this)f(case,)125 3452 y(these)j(c)m(haracter)i (v)-5 b(arieties)33 b(are)h(one)f(of)h(the)f(three)h(incarnations)e(of) h(the)g(mo)s(duli)e(space)j(of)f(Higgs)125 3560 y(bundles,)g(as)i (stated)h(b)m(y)f(the)g(celebrated)g(non-ab)s(elian)e(Ho)s(dge)i (corresp)s(ondence)f([)3062 3560 y SDict begin H.S end 3062 3560 a 0 TeXcolorgray FK(1)p 0 TeXcolorgray 3108 3502 a SDict begin H.R end 3108 3502 a 3108 3560 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Corlette:1988) cvn H.B /ANN pdfmark end 3108 3560 a FK(,)3168 3560 y SDict begin H.S end 3168 3560 a 0 TeXcolorgray FK(17)p 0 TeXcolorgray 3259 3502 a SDict begin H.R end 3259 3502 a 3259 3560 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.hitchin) cvn H.B /ANN pdfmark end 3259 3560 a FK(,)3319 3560 y SDict begin H.S end 3319 3560 a 0 TeXcolorgray FK(31)p 0 TeXcolorgray 3410 3502 a SDict begin H.R end 3410 3502 a 3410 3560 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.SimpsonI) cvn H.B /ANN pdfmark end 3410 3560 a FK(,)3470 3560 y SDict begin H.S end 3470 3560 a 0 TeXcolorgray FK(32)p 0 TeXcolorgray 3561 3502 a SDict begin H.R end 3561 3502 a 3561 3560 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.SimpsonII) cvn H.B /ANN pdfmark end 3561 3560 a FK(].)125 3668 y(F)-8 b(or)32 b(this)e(reason,)i(c)m (haracter)h(v)-5 b(arieties)31 b(of)h(surface)f(groups)g(ha)m(v)m(e)i (b)s(een)d(widely)g(studied,)g(particu-)125 3776 y(larly)25 b(regarding)i(the)g(computation)g(of)g(some)h(algebraic)f(in)m(v)-5 b(arian)m(ts)26 b(lik)m(e)h(their)f FD(E)5 b FK(-p)s(olynomial)25 b(\(an)125 3884 y(alternating)h(sum)h(of)g(its)g(Ho)s(dge)g(n)m(um)m(b) s(ers)f(in)g(the)i(spirit)d(of)i(the)g(Euler)f(c)m(haracteristic\).)41 b(Comput-)125 3992 y(ing)26 b(suc)m(h)i(in)m(v)-5 b(arian)m(t)26 b(is)h(a)h(hard)f(problem)f(that)i(has)f(b)s(een)g(tac)m(kled)i(from)e (an)g(arithmetic)g(viewp)s(oin)m(t)125 4100 y([)150 4100 y SDict begin H.S end 150 4100 a 0 TeXcolorgray FK(14)p 0 TeXcolorgray 241 4041 a SDict begin H.R end 241 4041 a 241 4100 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Hausel-Letellier-Villegas:2013) cvn H.B /ANN pdfmark end 241 4100 a FK(,)295 4100 y SDict begin H.S end 295 4100 a 0 TeXcolorgray FK(15)p 0 TeXcolorgray 386 4041 a SDict begin H.R end 386 4041 a 386 4100 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Hausel-Rodriguez-Villegas:2008) cvn H.B /ANN pdfmark end 386 4100 a FK(],)j(from)g(a)g(geometric)g (p)s(ersp)s(ectiv)m(e)f([)1669 4100 y SDict begin H.S end 1669 4100 a 0 TeXcolorgray FK(22)p 0 TeXcolorgray 1760 4041 a SDict begin H.R end 1760 4041 a 1760 4100 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.lomune) cvn H.B /ANN pdfmark end 1760 4100 a FK(,)1815 4100 y SDict begin H.S end 1815 4100 a 0 TeXcolorgray FK(26)p 0 TeXcolorgray 1906 4041 a SDict begin H.R end 1906 4041 a 1906 4100 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MM) cvn H.B /ANN pdfmark end 1906 4100 a FK(])g(and)g(from)g(the)h(p)s(oin)m(t)f(of)g(view)g(of)h (T)-8 b(op)s(ological)125 4208 y(Quan)m(tum)29 b(Field)g(Theories)h([) 1157 4208 y SDict begin H.S end 1157 4208 a 0 TeXcolorgray FK(9)p 0 TeXcolorgray 1202 4149 a SDict begin H.R end 1202 4149 a 1202 4208 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.GP-2019) cvn H.B /ANN pdfmark end 1202 4208 a FK(,)1258 4208 y SDict begin H.S end 1258 4208 a 0 TeXcolorgray FK(10)p 0 TeXcolorgray 1349 4149 a SDict begin H.R end 1349 4149 a 1349 4208 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.GPLM-2017) cvn H.B /ANN pdfmark end 1349 4208 a FK(].)224 4351 y(Character)f(v)-5 b(arieties)27 b(also)i(pla)m(y)e(a)i(prominen)m (t)e(role)h(in)f(the)h(top)s(ology)h(of)f(3-manifolds,)g(starting)125 4459 y(with)d(the)j(foundational)d(w)m(ork)i(of)g(Culler)e(and)i 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b(of)f(v)m(ery)h(in)m(tense)f (researc)m(h.)46 b(In)32 b(particular,)f(they)i(ha)m(v)m(e)g(b)s(een)f (used)f(to)i(study)e(knots)i FD(K)i FC(\032)125 5142 y FD(S)186 5109 y FJ(3)225 5142 y FK(,)42 b(b)m(y)e(analyzing)e(the)i (c)m(haracter)i(v)-5 b(arieties)39 b(asso)s(ciated)h(to)g(the)g (fundamen)m(tal)e(group)i(of)f(their)p 0 TeXcolorgray 125 5230 499 4 v 224 5322 a FI(2020)29 b Fy(Mathematics)g(Subje)l(ct)g (Classi\014c)l(ation.)39 b FI(Primary:)34 b(14M35.)i(Secondary:)e (57K31,)28 b(14D20,)f(14C15.)224 5413 y Fy(Key)h(wor)l(ds)h(and)f(phr)l (ases.)40 b FI(T)-6 b(orus)26 b(knot,)f(c)n(haracter)h(v)l(arieties,)h (represen)n(tations.)p 0 TeXcolorgray 0 TeXcolorgray 1850 5513 a FJ(1)p 0 TeXcolorgray eop end end %%Page: 2 2 TeXDict begin HPSdict begin 2 1 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show 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1>>/Subtype /Link/Dest (cite.Munoz) cvn H.B /ANN pdfmark end 650 996 a FK(],)j(and)e(the)h(case)h FD(G)c FK(=)g(SL)1582 1010 y FJ(3)1621 996 y FK(\()p Fz(C)18 b FK(\))40 b(w)m(as)35 b(addressed)d(in)h([)2523 996 y SDict begin H.S end 2523 996 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 2614 937 a SDict begin H.R end 2614 937 a 2614 996 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 2614 996 a FK(].)51 b(It)34 b(w)m(as)g(also)g(studied)e(for)125 1104 y FD(G)25 b FK(=)g(SU\(2\))31 b(in)e([)713 1104 y SDict begin H.S end 713 1104 a 0 TeXcolorgray FK(25)p 0 TeXcolorgray 805 1045 a SDict begin H.R end 805 1045 a 805 1104 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Martinez-Munoz:2015) cvn H.B /ANN pdfmark end 805 1104 a FK(].)224 1247 y(Nev)m(ertheless,)k(m)m(uc)m(h) e(less)g(is)g(kno)m(wn)g(in)f(the)i(higher)e(rank)h(case)h FD(r)e FC(\025)d FK(4,)32 b(ev)m(en)h(for)e(torus)g(knots.)125 1355 y(In)23 b(particular,)h(pro)m(viding)d(a)k(complete)f(description) e(of)i(their)f(geometry)j(is)d(still)f(an)h(op)s(en)h(problem.)125 1463 y(In)k(this)h(pap)s(er,)g(w)m(e)h(will)d(address)i(the)h(problem)e (of)i(computing,)f(o)m(v)m(er)i(an)e(arbitrary)g(algebraically)125 1571 y(closed)h(\014eld)f FD(k)k FK(of)e(zero)g(c)m(haracteristic,)g (the)f(virtual)f(classes)h(of)h(the)f(c)m(haracter)i(v)-5 b(arieties)30 b(of)h(torus)125 1679 y(knots)38 b(in)g(the)h (Grothendiec)m(k)g(ring)e(of)i(algebraic)g(v)-5 b(arieties,)41 b FD(K)7 b FK(\()p FC(V)g FD(ar)2592 1694 y Fx(k)2635 1679 y FK(\).)66 b(As)39 b(w)m(e)g(will)d(see,)42 b(these)125 1787 y(classes)28 b(lie)f(in)f(the)j(subring)d(of)i FD(K)7 b FK(\()p FC(V)g FD(ar)1483 1802 y Fx(k)1526 1787 y FK(\))28 b(generated)h(b)m(y)f(the)h(class)f(of)g(the)g(a\016ne)h(line)d FD(q)i FK(=)d([)p FD(k)s FK(])k(\(the)125 1895 y(so-called)j(Lefsc)m (hetz)i(motiv)m(e\),)h(so)d(it)h(actually)f(computes)h(the)g(motiv)m(e) g(of)g(the)g(c)m(haracter)h(v)-5 b(ariet)m(y)125 2003 y(in)29 b(the)h(Cho)m(w)g(ring)f(and)h(its)g FD(E)5 b FK(-p)s(olynomial.)224 2146 y(This)21 b(pap)s(er)h(is)g(structured)g (as)h(follo)m(ws.)37 b(In)22 b(Section)2041 2146 y SDict begin H.S end 2041 2146 a 0 TeXcolorgray FK(2)p 0 TeXcolorgray 2087 2088 a SDict begin H.R end 2087 2088 a 2087 2146 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.2) cvn H.B /ANN pdfmark end 2087 2146 a 22 w FK(w)m(e)i(review)e(some)h (generalities)f(ab)s(out)h(rep-)125 2254 y(resen)m(tation)i(and)g(c)m (haracter)h(v)-5 b(arieties)25 b(as)g(w)m(ell)f(as)h(the)h(Grothendiec) m(k)f(ring)f(of)h(algebraic)g(v)-5 b(arieties.)125 2362 y(P)m(articularly)d(,)38 b(in)e(Section)1091 2362 y SDict begin H.S end 1091 2362 a 0 TeXcolorgray FK(2.1)p 0 TeXcolorgray 1208 2303 a SDict begin H.R end 1208 2303 a 1208 2362 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.2.1) cvn H.B /ANN pdfmark end 1208 2362 a 37 w FK(w)m(e)i(in)m(tro)s(duce)e (a)i(canonical)f(\014ltration)f(attac)m(hed)k(to)e(a)g(represen-)125 2470 y(tation,)33 b(with)e(semi-simple)f(graded)i(represen)m(tation,)i (that)f(w)m(e)g(will)c(refer)j(to)i(as)e(the)h(semi-simple)125 2578 y(\014ltration.)42 b(This)30 b(\014ltration)g(will)f(b)s(e)i(v)m (ery)h(useful)e(b)s(ecause)h(it)g(induces)f(a)i(natural)f(decomp)s (osition)125 2686 y(of)40 b(the)h(represen)m(tation)g(v)-5 b(ariet)m(y)41 b(according)g(to)g(the)g(t)m(yp)s(e)g(\(i.e.)g(the)g (dimension)d(and)i(m)m(ultiplic-)125 2794 y(it)m(y)e(of)g(eac)m(h)h (isot)m(ypic)f(comp)s(onen)m(t)g(of)g(the)h(graded)f(pieces\))g(of)g (the)g(semi-simple)e(\014ltration)h(of)h(a)125 2902 y(represen)m (tation.)224 3045 y(In)32 b(Section)660 3045 y SDict begin H.S end 660 3045 a 0 TeXcolorgray FK(3)p 0 TeXcolorgray 706 2986 a SDict begin H.R end 706 2986 a 706 3045 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.3) cvn H.B /ANN pdfmark end 706 3045 a FK(,)h(w)m(e)g(sp)s(ecialize)e(to)j(the)e (case)i(of)f(torus)f(knots.)47 b(There,)33 b(w)m(e)g(outline)e(the)i (strategy)125 3153 y(that)d(w)m(e)g(will)d(follo)m(w)i(throughout)h (this)e(pap)s(er.)40 b(W)-8 b(e)31 b(fo)s(cus)e(on)g(the)h(sub)m(v)-5 b(ariet)m(y)30 b FD(R)3003 3120 y FJ(irr)3081 3153 y FK(\(\000)p FD(;)15 b FK(SL)3321 3167 y Fx(r)3359 3153 y FK(\()p FD(k)s FK(\)\))27 b FC(\032)125 3261 y FD(R)q FK(\(\000)p FD(;)15 b FK(SL)434 3275 y Fx(r)472 3261 y FK(\()p FD(k)s FK(\)\))36 b(of)g(irreducible)31 b(represen)m (tations.)55 b(The)34 b(k)m(ey)i(p)s(oin)m(t)e(is)g(that,)j(as)f(sho)m (wn)e(in)g(Prop)s(o-)125 3369 y(sition)380 3369 y SDict begin H.S end 380 3369 a 0 TeXcolorgray FK(3.2)p 0 TeXcolorgray 497 3310 a SDict begin H.R end 497 3310 a 497 3369 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.2) cvn H.B /ANN pdfmark end 497 3369 a FK(,)40 b(an)e(irreducible)d (represen)m(tation)k FD(\032)f FK(:)h(\000)1963 3383 y Fx(n;m)2130 3369 y FC(!)g FK(SL)2367 3383 y Fx(r)2405 3369 y FK(\()p FD(k)s FK(\))g(lifts,)g(up)e(to)i(rescaling,)h(to)f(a) 125 3477 y(represen)m(tation)30 b(of)h(the)f(free)h(pro)s(duct)1435 3634 y Fz(Z)1500 3648 y Fx(n)1563 3634 y FD(?)21 b Fz(Z)1694 3648 y Fx(m)1891 3740 y FJ(~)-42 b Fx(\032)2035 3877 y Fw(\045)p Fv(\045)2000 3855 y Fu(L)1925 3810 y(L)1850 3765 y(L)1775 3719 y(L)1700 3674 y(L)1503 3972 y FK(\000)1560 3986 y Fx(n;m)1857 4005 y(\032)1982 3950 y Fw(/)p Fv(/)p 1716 3951 267 4 v 1596 3674 a Fw(O)p Fv(O)p 1594 3883 4 209 v 2009 3972 a FK(SL)2117 3986 y Fx(r)2154 3972 y FK(\()p FD(k)s FK(\))p FD(:)125 4158 y FK(This)25 b(op)s(ens)i(the)h (do)s(or)f(to)h(computing)f(the)h(motiv)m(e)g(of)g(the)f(irreducible)e (represen)m(tation)i(v)-5 b(ariet)m(y)28 b(of)125 4266 y(torus)i(knots)g(through)g(the)g(represen)m(tation)h(v)-5 b(ariet)m(y)31 b(of)f Fz(Z)2167 4280 y Fx(n)2230 4266 y FD(?)20 b Fz(Z)2360 4280 y Fx(m)2453 4266 y FK(via)489 4417 y([)p FD(R)584 4379 y FJ(irr)663 4417 y FK(\(\000)755 4431 y Fx(n;m)884 4417 y FD(;)15 b FK(SL)1032 4431 y Fx(r)1070 4417 y FK(\()p FD(k)s FK(\)\)])26 b(=)f([)p FD(R)q FK(\()p Fz(Z)1567 4431 y Fx(n)1630 4417 y FD(?)c Fz(Z)1761 4431 y Fx(m)1823 4417 y FD(;)15 b FK(SL)1971 4431 y Fx(r)2009 4417 y FK(\()p FD(k)s FK(\)\)])22 b FC(\000)d FK([)p FD(R)2396 4379 y FJ(red)2499 4417 y FK(\()p Fz(Z)2599 4431 y Fx(n)2662 4417 y FD(?)h Fz(Z)2792 4431 y Fx(m)2855 4417 y FD(;)15 b FK(SL)3002 4431 y Fx(r)3040 4417 y FK(\()p FD(k)s FK(\)\)])p FD(;)125 4572 y FK(where)29 b FD(R)457 4539 y FJ(red)559 4572 y FK(\()p Fz(Z)659 4586 y Fx(n)722 4572 y FD(?)21 b Fz(Z)853 4586 y Fx(m)915 4572 y FD(;)15 b FK(SL)1063 4586 y Fx(r)1101 4572 y FK(\()p FD(k)s FK(\)\))32 b(is)d(the)i(sub)m(v)-5 b(ariet)m(y)30 b(of)g(reducible)e(represen)m(tations)j(of)f Fz(Z)3262 4586 y Fx(n)3325 4572 y FD(?)21 b Fz(Z)3456 4586 y Fx(m)3518 4572 y FK(.)224 4716 y(This)28 b(greatly)j(simpli\014es)26 b(the)k(problem)f(b)s(ecause)g(the)i(represen)m(tations)e FD(\032)d FK(:)f Fz(Z)2950 4730 y Fx(n)3012 4716 y FD(?)20 b Fz(Z)3142 4730 y Fx(m)3229 4716 y FC(!)25 b FK(SL)3453 4730 y Fx(r)3491 4716 y FK(\()p FD(k)s FK(\))125 4823 y(retain)34 b(some)h(crucial)f(prop)s(erties)f(from)i(the)g(classical)f (represen)m(tation)h(theory)g(of)g(\014nite)f(groups,)125 4931 y(for)25 b(instance,)h(only)f(\014nitely)f(man)m(y)i (con\014gurations)e(of)i(eigen)m(v)-5 b(alues)25 b(are)h(allo)m(w)m(ed) g(for)f(the)h(elemen)m(ts)125 5039 y FD(\032)p FK(\()p FD(x)p FK(\))p FD(;)15 b(\032)p FK(\()p FD(y)s FK(\))26 b FC(2)f FK(SL)719 5053 y Fx(r)757 5039 y FK(\()p FD(k)s FK(\).)39 b(Hence,)27 b(w)m(e)d(can)h(re\014ne)e(the)i(decomp)s (osition)e(of)h(the)h(represen)m(tation)f(v)-5 b(ariet)m(y)125 5147 y(according)30 b(to)h(its)f(semi-simple)e(\014ltration)h(in)g (order)h(to)h(obtain)f(a)g(decomp)s(osition)125 5310 y(\(1\))1110 5310 y SDict begin H.S end 1110 5310 a 1110 5310 a SDict begin 13 H.A end 1110 5310 a 1110 5310 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.1) cvn /DEST pdfmark end 1110 5310 a FD(R)1180 5272 y FJ(red)1282 5310 y FK(\()p Fz(Z)1382 5324 y Fx(n)1445 5310 y FD(?)21 b Fz(Z)1576 5324 y Fx(m)1638 5310 y FD(;)15 b FK(SL)1786 5324 y Fx(r)1823 5310 y FK(\()p FD(k)s FK(\)\))27 b(=)2101 5224 y Ft(G)2131 5415 y Fx(\024)2256 5224 y Ft(G)2217 5421 y Fx(\034)8 b Fs(2T)2359 5402 y Fr(\003)2342 5437 y Fq(\024)2410 5310 y FD(R)q FK(\()p FD(\034)i FK(\))p FD(;)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.3) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)560 b(3)p 0 TeXcolorgray 125 266 a FK(where)33 b FD(\024)i FK(runs)e(o)m(v)m(er)j(the)e(p)s(ossible) e(con\014gurations)i(of)h(eigen)m(v)-5 b(alues,)35 b FC(T)2662 233 y Fs(\003)2639 288 y Fx(\024)2735 266 y FK(are)g(the)g(t)m(yp)s(es)f(of)h(semi-)125 374 y(simple)k (\014ltrations)i(that)h(corresp)s(ond)f(to)i(reducible)c(represen)m (tations)j(with)f(eigen)m(v)-5 b(alues)41 b(tak)m(en)125 482 y(from)34 b FD(\024)p FK(,)i(and)e FD(R)q FK(\()p FD(\034)10 b FK(\))36 b(is)d(the)i(collection)g(of)g(represen)m (tations)g FD(\032)d FK(:)h Fz(Z)2474 496 y Fx(n)2540 482 y FD(?)23 b Fz(Z)2673 496 y Fx(m)2768 482 y FC(!)33 b FK(SL)2999 496 y Fx(r)3037 482 y FK(\()p FD(k)s FK(\))i(with)f(semi-) 125 589 y(simple)i(\014ltration)i(of)h(t)m(yp)s(e)g FD(\034)50 b FC(2)39 b(T)1392 556 y Fs(\003)1369 612 y Fx(\024)1431 589 y FK(.)67 b(Notice)40 b(that)f(the)g(previous)f(decomp)s(osition)f (has)i(\014nitely)125 697 y(man)m(y)30 b(strata)h(since)f(there)h(are)f (\014nitely)f(man)m(y)h(c)m(hoices)i(for)e FD(\024)g FK(and)g FD(\034)10 b FK(.)224 841 y(Lea)m(ving)21 b(aside)f(for)h(a)g (momen)m(t)g(the)g(coun)m(t)g(of)g(the)g(n)m(um)m(b)s(er)e(of)i(con)m (tributing)e(strata,)24 b(the)d(problem)125 949 y(of)32 b(computing)f(the)h(motiv)m(e)g([)p FD(R)q FK(\()p FD(\034)10 b FK(\)])29 b FC(2)e FD(K)7 b FK(\()p FC(V)g FD(ar)1766 964 y Fx(k)1809 949 y FK(\))33 b(for)e(a)h(\014xed)g(t)m(yp)s(e)g FD(\034)42 b FK(is)31 b(addressed)f(in)h(Section)3541 949 y SDict begin H.S end 3541 949 a 0 TeXcolorgray FK(4)p 0 TeXcolorgray 3586 890 a SDict begin H.R end 3586 890 a 3586 949 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.4) cvn H.B /ANN pdfmark end 3586 949 a FK(,)125 1057 y(whic)m(h)26 b(is)i(the)g(heart)g(of)h(this)e(pap)s(er.)39 b(Roughly)27 b(sp)s(eaking,)h(in)e(Prop)s(osition)2777 1057 y SDict begin H.S end 2777 1057 a 0 TeXcolorgray FK(4.5)p 0 TeXcolorgray 2893 998 a SDict begin H.R end 2893 998 a 2893 1057 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.5) cvn H.B /ANN pdfmark end 2893 1057 a 28 w FK(and)h(Corollary) 3495 1057 y SDict begin H.S end 3495 1057 a 0 TeXcolorgray FK(4.7)p 0 TeXcolorgray 3611 998 a SDict begin H.R end 3611 998 a 3611 1057 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.7) cvn H.B /ANN pdfmark end 3611 1057 a 125 1165 a FK(w)m(e)34 b(will)e(sho)m(w)i(that)g(\(for)h(lo)m(w)f(rank\))g FD(R)q FK(\()p FD(\034)10 b FK(\))34 b(is)f(the)i(total)g(space)f(of)h (a)f(lo)s(cally)f(trivial)f(\014bration)h(in)125 1272 y(the)d(Zariski)e(top)s(ology)125 1489 y(\(2\))1620 1489 y SDict begin H.S end 1620 1489 a 1620 1489 a SDict begin 13 H.A end 1620 1489 a 1620 1489 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.2) cvn /DEST pdfmark end 1620 1489 a FD(R)q FK(\()p FD(\034)10 b FK(\))26 b FC(!)f FB(M)2047 1503 y Fx(\034)2091 1489 y FD(;)125 1705 y FK(that)35 b(\014b)s(ers)f(o)m(v)m(er)j(the)e(sub)m(v)-5 b(ariet)m(y)35 b FB(M)1473 1719 y Fx(\034)1552 1705 y FK(of)h(semi-simple)c(represen)m(tations)k(of)f(t)m(yp)s(e)h FD(\034)10 b FK(.)55 b(The)35 b(\014b)s(er)125 1813 y(of)k(this)g(map)h (is)e(giv)m(en)i(b)m(y)g(a)g(quotien)m(t)g FC(M)1681 1827 y Fx(\034)1724 1813 y FD(=)p FC(G)1823 1827 y Fx(\034)1867 1813 y FK(,)i(where)d FC(M)2315 1827 y Fx(\034)2398 1813 y FK(is)g(a)h(v)-5 b(ariet)m(y)40 b(parametrizing)f(the)125 1921 y(w)m(a)m(ys)33 b(in)f(whic)m(h)g(a)h(semi-simple)e(represen)m (tation)i(can)g(b)s(e)f(completed)h(to)h(an)f(arbitrary)f(reducible)125 2029 y(represen)m(tation)37 b(and)f FC(G)963 2043 y Fx(\034)1043 2029 y FC(\032)g FK(GL)1278 2043 y Fx(r)1316 2029 y FK(\()p FD(k)s FK(\))i(is)e(a)h Fp(gauge)i(gr)-5 b(oup)43 b FK(acting)38 b(freely)e(on)h FC(M)2949 2043 y Fx(\034)2992 2029 y FK(,)i(that)f(iden)m(ti\014es)125 2137 y(equiv)-5 b(alen)m(t)29 b(completions.)224 2280 y(The)f(description)e(of)i(Section)1289 2280 y SDict begin H.S end 1289 2280 a 0 TeXcolorgray FK(4)p 0 TeXcolorgray 1334 2222 a SDict begin H.R end 1334 2222 a 1334 2280 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.4) cvn H.B /ANN pdfmark end 1334 2280 a 28 w FK(will)d(b)s(e)j(enough)g(for)f(computing)g(the)i(motiv)m(es)f(of)g FC(M)3243 2294 y Fx(\034)3315 2280 y FK(and)f FC(G)3543 2294 y Fx(\034)3586 2280 y FK(.)125 2388 y(In)i(this)g(w)m(a)m(y)-8 b(,)32 b(in)c(Section)1040 2388 y SDict begin H.S end 1040 2388 a 0 TeXcolorgray FK(5)p 0 TeXcolorgray 1086 2330 a SDict begin H.R end 1086 2330 a 1086 2388 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.5) cvn H.B /ANN pdfmark end 1086 2388 a 30 w FK(w)m(e)i(giv)m(e)h(a)f(com)m(binatorial)f(recip)s (e)g(for)h(calculating)g(them)g(in)e(terms)i(of)125 2496 y(the)35 b(rep)s(eated)g(eigen)m(v)-5 b(alues)35 b(of)g FD(\024)g FK(and)f(the)h(t)m(yp)s(e)h FD(\034)10 b FK(.)54 b(Observ)m(e)35 b(that,)i(since)d(w)m(e)i(are)f(dealing)f(with)125 2604 y(reducible)39 b(represen)m(tations,)46 b(all)41 b(the)h(irreducible)d(comp)s(onen)m(ts)j(app)s(earing)e(in)h FB(M)3122 2618 y Fx(\034)3208 2604 y FK(are)h(forced)125 2712 y(to)j(ha)m(v)m(e)h(lo)m(w)m(er)f(rank.)83 b(In)44 b(suc)m(h)g(manner,)k(the)d(results)e(of)i(this)e(section)i(pro)m(vide) f(a)h(recursiv)m(e)125 2820 y(metho)s(d)c(for)g(computing)g(the)h (motiv)m(e)g(of)g FD(R)q FK(\()p FD(\034)10 b FK(\))44 b FC(\032)g FD(R)q FK(\(\000)2181 2834 y Fx(n;m)2310 2820 y FD(;)15 b FK(SL)2458 2834 y Fx(r)2496 2820 y FK(\()p FD(k)s FK(\)\))43 b(from)e(the)h(kno)m(wledge)f(of)125 2932 y([)p FD(R)220 2899 y FJ(irr)298 2932 y FK(\(\000)390 2946 y Fx(n;m)519 2932 y FD(;)15 b FK(SL)667 2946 y Fx(s)704 2932 y FK(\()p FD(k)s FK(\)\)])32 b(for)e FD(s)25 b(<)g(r)s FK(.)224 3076 y(Regarding)41 b(the)h(n)m(um)m(b)s(er)e(of)h(comp)s (onen)m(ts)h(in)e(\()1959 3076 y SDict begin H.S end 1959 3076 a 0 TeXcolorgray FK(1)p 0 TeXcolorgray 2005 3017 a SDict begin H.R end 2005 3017 a 2005 3076 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.1) cvn H.B /ANN pdfmark end 2005 3076 a FK(\),)k(as)e(a)g(b)m(ypro)s(duct)e(of)h(Section)3210 3076 y SDict begin H.S end 3210 3076 a 0 TeXcolorgray FK(5)p 0 TeXcolorgray 3256 3017 a SDict begin H.R end 3256 3017 a 3256 3076 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.5) cvn H.B /ANN pdfmark end 3256 3076 a FK(,)j(w)m(e)e(will)125 3184 y(sho)m(w)30 b(that)i(the)f(motiv)m(e)h([)p FD(R)q FK(\()p FD(\034)10 b FK(\)])32 b(for)e FD(\034)37 b FC(2)25 b(T)1651 3151 y Fs(\003)1628 3206 y Fx(\024)1721 3184 y FK(do)s(es)31 b(not)g(dep)s(end)e(on)i(the)g(particular)f(v)-5 b(alues)30 b(of)h(the)125 3292 y(eigen)m(v)-5 b(alues)22 b(of)g(the)g(con\014guration)g FD(\024)p FK(,)i(but)d(only)h(on)g(the)g (m)m(ultiplicit)m(y)d(of)k(its)e(rep)s(eated)h(eigen)m(v)-5 b(alues.)125 3399 y(Hence,)31 b(the)g(sum)e(in)g FD(\024)i FK(of)f(\()1096 3399 y SDict begin H.S end 1096 3399 a 0 TeXcolorgray FK(1)p 0 TeXcolorgray 1142 3341 a SDict begin H.R end 1142 3341 a 1142 3399 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.1) cvn H.B /ANN pdfmark end 1142 3399 a FK(\))h(amoun)m(ts)g(to)g(m)m(ultiplying)c([)p FD(R)q FK(\()p FD(\034)10 b FK(\)])31 b(b)m(y)f(a)h(com)m(binatorial)f (co)s(e\016cien)m(t)125 3507 y(that)f(coun)m(ts)g(the)g(n)m(um)m(b)s (er)f(of)g(admissible)e(con\014gurations)i(with)g(prescrib)s(ed)e(m)m (ultiplicities.)36 b(This)125 3615 y(coun)m(ting)e(is)g(a)i(com)m (binatorial)e(problem)f(that)j(is)e(addressed)g(in)g(Section)2764 3615 y SDict begin H.S end 2764 3615 a 0 TeXcolorgray FK(6)p 0 TeXcolorgray 2810 3557 a SDict begin H.R end 2810 3557 a 2810 3615 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.6) cvn H.B /ANN pdfmark end 2810 3615 a FK(.)54 b(In)35 b(Section)3330 3615 y SDict begin H.S end 3330 3615 a 0 TeXcolorgray FK(6.5)p 0 TeXcolorgray 3447 3557 a SDict begin H.R end 3447 3557 a 3447 3615 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.6.5) cvn H.B /ANN pdfmark end 3447 3615 a FK(,)h(w)m(e)125 3723 y(p)s(erform)42 b(the)j(calculation)e(in)g(the)h(case)i(that)e(the)h(\(rather)f (arti\014cial\))f(condition)g(gcd\()p FD(n;)15 b(r)s FK(\))49 b(=)125 3831 y(gcd\()p FD(m;)15 b(r)s FK(\))26 b(=)f(1)i(holds,)f(obtaining)f(closed)h(form)m(ulas)g(for)g(these)h(co) s(e\016cien)m(ts)h(in)d(terms)h(of)h(the)g(usual)125 3939 y(m)m(ultinomial)40 b(n)m(um)m(b)s(ers.)79 b(This)42 b(h)m(yp)s(othesis)g(is)h(dropp)s(ed)f(in)g(Section)2694 3939 y SDict begin H.S end 2694 3939 a 0 TeXcolorgray FK(6.4)p 0 TeXcolorgray 2810 3881 a SDict begin H.R end 2810 3881 a 2810 3939 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.6.4) cvn H.B /ANN pdfmark end 2810 3939 a FK(,)47 b(where)d(the)f(general)125 4047 y(case)33 b(is)e(studied)g(b)m(y)h (means)g(of)h(generating)f(functions.)45 b(In)32 b(that)h(setting,)g(w) m(e)g(get)g(more)g(in)m(v)m(olv)m(ed)125 4155 y(form)m(ulas)f(that)j (turn)e(out)h(to)g(agree)h(with)e(the)h(simpler)d(closed)j(forms)f(of)h (Section)3072 4155 y SDict begin H.S end 3072 4155 a 0 TeXcolorgray FK(6.5)p 0 TeXcolorgray 3188 4096 a SDict begin H.R end 3188 4096 a 3188 4155 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.6.5) cvn H.B /ANN pdfmark end 3188 4155 a 34 w FK(for)f FD(r)h FC(\024)d FK(4,)125 4263 y(ev)m(en)g(if)e(the)i(primalit)m(y)d(condition)h(no)h(longer)g(holds.) 224 4406 y(In)45 b(Section)686 4406 y SDict begin H.S end 686 4406 a 0 TeXcolorgray FK(7)p 0 TeXcolorgray 731 4348 a SDict begin H.R end 731 4348 a 731 4406 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.7) cvn H.B /ANN pdfmark end 731 4406 a FK(,)k(w)m(e)d(use)f(the)g(metho)s(d)g(dev)m(elop)s(ed)g (in)f(this)g(pap)s(er)g(to)i(compute)f(the)h(motiv)m(e)125 4514 y(of)41 b(the)h(c)m(haracter)i(v)-5 b(ariet)m(y)42 b(of)g(irreducible)c(represen)m(tations)k(of)g(torus)g(knots)f(for)h FD(G)j FK(=)f(SL)3451 4528 y FJ(2)3491 4514 y FK(\()p FD(k)s FK(\))125 4622 y(\(Section)483 4622 y SDict begin H.S end 483 4622 a 0 TeXcolorgray FK(7.1)p 0 TeXcolorgray 599 4564 a SDict begin H.R end 599 4564 a 599 4622 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.7.1) cvn H.B /ANN pdfmark end 599 4622 a FK(\))36 b(and)f FD(G)g FK(=)e(SL)1169 4636 y FJ(3)1209 4622 y FK(\()p FD(k)s FK(\))j(\(Section)1724 4622 y SDict begin H.S end 1724 4622 a 0 TeXcolorgray FK(7.2)p 0 TeXcolorgray 1840 4564 a SDict begin H.R end 1840 4564 a 1840 4622 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.7.2) cvn H.B /ANN pdfmark end 1840 4622 a FK(\).)57 b(This)34 b(repro)m(v)m(es)i(the)g (existing)e(calculations)h(in)125 4730 y(the)25 b(literature)f(of)h (these)h(motiv)m(es)f(in)f([)1450 4730 y SDict begin H.S end 1450 4730 a 0 TeXcolorgray FK(27)p 0 TeXcolorgray 1541 4671 a SDict begin H.R end 1541 4671 a 1541 4730 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Munoz) cvn H.B /ANN pdfmark end 1541 4730 a FK(])i(and)e([)1788 4730 y SDict begin H.S end 1788 4730 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 1879 4671 a SDict begin H.R end 1879 4671 a 1879 4730 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 1879 4730 a FK(],)j(resp)s(ectiv)m(ely) -8 b(.)39 b(F)-8 b(or)25 b(that)h(purp)s(ose,)f(w)m(e)g(pro)m(vide)125 4838 y(a)35 b(detailed)g(description)e(of)i(the)h(p)s(ossible)d(t)m(yp) s(es)i(and)g(con\014gurations)f(of)i(eigen)m(v)-5 b(alues)35 b(that)h(ma)m(y)125 4946 y(o)s(ccur.)224 5089 y(The)g(aim)f(of)h (Section)1032 5089 y SDict begin H.S end 1032 5089 a 0 TeXcolorgray FK(8)p 0 TeXcolorgray 1077 5031 a SDict begin H.R end 1077 5031 a 1077 5089 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.8) cvn H.B /ANN pdfmark end 1077 5089 a 36 w FK(is)f(to)i(address)e(the)h(computation)g(of)g(the)g(c)m (haracter)i(v)-5 b(ariet)m(y)36 b(of)g(irre-)125 5197 y(ducible)25 b(represen)m(tations)i(in)f(the)h(case)i FD(G)c FK(=)g(SL)1808 5211 y FJ(4)1847 5197 y FK(\()p FD(k)s FK(\),)k FB(M)2116 5164 y FJ(irr)2195 5197 y FK(\(\000)p FD(;)15 b FK(SL)2435 5211 y FJ(4)2475 5197 y FK(\()p FD(k)s FK(\)\),)29 b(whic)m(h)d(is)g(the)i(main)e(no)m(v)m(el)125 5305 y(con)m(tribution)37 b(of)j(this)e(pap)s(er.)66 b(This)38 b(rank)h(is)f(the)h(\014rst)g(case)h(in)e(whic)m(h)g(w)m(e)i (need)f(to)h(deal)f(with)125 5413 y(represen)m(tations)28 b(with)f(isot)m(ypic)i(comp)s(onen)m(ts)f(of)h(dimension)d(2)j(and)f(m) m(ultiplicit)m(y)e(higher)h(than)i(1.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View 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end 1560 949 a 34 w FK(since,)i(roughly)d(sp)s(eaking,)i FC(M)2694 963 y Fx(\034)2772 949 y FK(is)f(the)h(complemen)m(t)g(of)125 1057 y(some)c(Sc)m(h)m(ub)s(ert)f(cells)h(as)g(describ)s(ed)e(in)h (Section)1868 1057 y SDict begin H.S end 1868 1057 a 0 TeXcolorgray FK(4)p 0 TeXcolorgray 1913 998 a SDict begin H.R end 1913 998 a 1913 1057 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.4) cvn H.B /ANN pdfmark end 1913 1057 a FK(.)41 b(Nonetheless,)30 b(this)f(more)i(in)m(v)m(olv)m(ed)e (situation)125 1165 y(can)40 b(b)s(e)g(analyzed)g(with)f(the)i(to)s (ols)f(in)m(tro)s(duced)e(in)h(Section)2344 1165 y SDict begin H.S end 2344 1165 a 0 TeXcolorgray FK(5)p 0 TeXcolorgray 2389 1106 a SDict begin H.R end 2389 1106 a 2389 1165 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.5) cvn H.B /ANN pdfmark end 2389 1165 a FK(.)70 b(Due)41 b(to)g(the)g(h)m(uge)f(n)m(um)m(b)s(er)f(of)125 1272 y(p)s(ossible)32 b(t)m(yp)s(es)j(that)h(w)m(e)g(need)e(to)i(consider)e (\(more)i(than)f(350)h(in)e(comparison)g(with)g(2)h(for)g FD(G)f FK(=)125 1380 y(SL)232 1394 y FJ(2)271 1380 y FK(\()p FD(k)s FK(\))e(and)e(23)h(for)f FD(G)c FK(=)f(SL)1161 1394 y FJ(3)1200 1380 y FK(\()p FD(k)s FK(\)\),)32 b(w)m(e)f(do)g(not)f (pro)m(vide)g(an)g(exhaustiv)m(e)h(en)m(umeration)f(of)h(all)e(the)125 1488 y(p)s(ossibilities.)38 b(Nev)m(ertheless,)33 b(the)e(description)e (of)j(Sections)2284 1488 y SDict begin H.S end 2284 1488 a 0 TeXcolorgray FK(4)p 0 TeXcolorgray 2330 1430 a SDict begin H.R end 2330 1430 a 2330 1488 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.4) cvn H.B /ANN pdfmark end 2330 1488 a 31 w FK(and)2539 1488 y SDict begin H.S end 2539 1488 a 0 TeXcolorgray FK(5)p 0 TeXcolorgray 2584 1430 a SDict begin H.R end 2584 1430 a 2584 1488 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.5) cvn H.B /ANN pdfmark end 2584 1488 a 31 w FK(is)e(explicit)g(enough)h(to)h(giv)m(e)125 1596 y(rise)f(to)i(a)g(precise)f(algorithm)g(that)h(can)g(b)s(e)e (implemen)m(ted)g(in)h(a)h(computer)f(algebra)g(system.)48 b(An)125 1704 y(implemen)m(tation)32 b(in)h(SageMath)j([)1343 1704 y SDict begin H.S end 1343 1704 a 0 TeXcolorgray FK(30)p 0 TeXcolorgray 1434 1646 a SDict begin H.R end 1434 1646 a 1434 1704 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.SageMath) cvn H.B /ANN pdfmark end 1434 1704 a FK(])e(created)i(b)m(y) e(the)g(authors)g(can)h(b)s(e)e(c)m(hec)m(k)m(ed)j(in)d([)3209 1704 y SDict begin H.S end 3209 1704 a 0 TeXcolorgray FK(11)p 0 TeXcolorgray 3300 1646 a SDict begin H.R end 3300 1646 a 3300 1704 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.GPMScript) cvn H.B /ANN pdfmark end 3300 1704 a FK(].)53 b(F)-8 b(rom)125 1812 y(this)29 b(computation,)h(w)m(e)h(obtain)f(the)h (main)e(result)g(of)i(this)e(pap)s(er.)125 1974 y SDict begin H.S end 125 1974 a 125 1974 a SDict begin 13 H.A end 125 1974 a 125 1974 a SDict begin [/View [/XYZ H.V]/Dest (theorem.1.1) cvn /DEST pdfmark end 125 1974 a 0 TeXcolorgray FL(Theorem)k(1.1.)p 0 TeXcolorgray 42 w Fp(The)f(motive)h(of)f(the)h(irr)-5 b(e)g(ducible)33 b FK(SL)2057 1988 y FJ(4)2097 1974 y FK(\()p FD(k)s FK(\))p Fp(-char)-5 b(acter)34 b(variety)f(of)g(the)f(gr)-5 b(oup)34 b FK(\000)25 b(=)125 2082 y(\000)182 2096 y Fx(n;m)343 2082 y Fp(of)33 b(the)g FK(\()p FD(n;)15 b(m)p FK(\))p Fp(-torus)33 b(knot)g(is)g(given)f(by)125 2238 y FK([)p FB(M)245 2200 y FJ(irr)324 2238 y FK(\(\000)p FD(;)15 b FK(SL)564 2252 y FJ(4)603 2238 y FK(\()p FD(k)s FK(\)\)])-5 b(=)905 2202 y FJ(4)p 871 2217 106 4 v 871 2269 a Fx(nm)986 2164 y Ft(\000)1027 2198 y Fx(n)1031 2269 y FJ(4)1070 2164 y Ft(\001\000)1154 2198 y Fx(m)1167 2269 y FJ(4)1216 2164 y Ft(\001\000)1299 2238 y FD(q)1343 2200 y FJ(9)1403 2238 y FK(+)20 b(6)p FD(q)1583 2200 y FJ(8)1643 2238 y FK(+)g(20)p FD(q)1868 2200 y FJ(7)1928 2238 y FK(+)g(17)p FD(q)2153 2200 y FJ(6)2213 2238 y FC(\000)g FK(98)p FD(q)2438 2200 y FJ(5)2499 2238 y FC(\000)f FK(26)p FD(q)2723 2200 y FJ(4)2784 2238 y FK(+)h(38)p FD(q)3009 2200 y FJ(3)3069 2238 y FK(+)g(126)p FD(q)3339 2200 y FJ(2)3400 2238 y FC(\000)g FK(144)3626 2164 y Ft(\001)380 2414 y FK(+)515 2378 y FJ(4)p 480 2393 V 480 2446 a Fx(nm)596 2313 y Ft(\020)650 2341 y(\000)692 2374 y Fx(n)695 2446 y FJ(4)734 2341 y Ft(\001\000)832 2374 y Fx(m)818 2446 y FJ(2)p Fx(;)p FJ(1)908 2341 y Ft(\001)970 2414 y FK(+)1061 2341 y Ft(\000)1126 2374 y Fx(n)1102 2446 y FJ(2)p Fx(;)p FJ(1)1192 2341 y Ft(\001\000)1276 2374 y Fx(m)1289 2446 y FJ(4)1338 2341 y Ft(\001)1380 2313 y(\021)1434 2341 y(\000)1476 2414 y FD(q)1520 2377 y FJ(7)1579 2414 y FK(+)g(5)p FD(q)1759 2377 y FJ(6)1819 2414 y FK(+)g(7)p FD(q)1999 2377 y FJ(5)2059 2414 y FC(\000)g FK(34)p FD(q)2284 2377 y FJ(4)2344 2414 y FK(+)g(34)p FD(q)2569 2377 y FJ(2)2630 2414 y FK(+)g(18)p FD(q)k FC(\000)19 b FK(48)3056 2341 y Ft(\001)380 2613 y FK(+)515 2577 y FJ(4)p 480 2592 V 480 2645 a Fx(nm)596 2512 y Ft(\020)650 2540 y(\000)692 2573 y Fx(n)695 2645 y FJ(4)734 2540 y Ft(\001\000)818 2573 y Fx(m)831 2645 y FJ(2)880 2540 y Ft(\001)942 2613 y FK(+)1033 2540 y Ft(\000)1075 2573 y Fx(n)1078 2645 y FJ(2)1117 2540 y Ft(\001\000)1201 2573 y Fx(m)1214 2645 y FJ(4)1263 2540 y Ft(\001)1305 2512 y(\021)1359 2540 y(\000)1401 2613 y FD(q)1445 2576 y FJ(5)1504 2613 y FK(+)h(4)p FD(q)1684 2576 y FJ(4)1744 2613 y FC(\000)g FK(11)p FD(q)1969 2576 y FJ(3)2029 2613 y FK(+)g FD(q)2164 2576 y FJ(2)2224 2613 y FK(+)g(18)p FD(q)k FC(\000)c FK(18)2651 2540 y Ft(\001)380 2812 y FK(+)515 2776 y FJ(4)p 480 2791 V 480 2843 a Fx(nm)596 2711 y Ft(\020)650 2738 y(\000)692 2772 y Fx(n)695 2843 y FJ(4)734 2738 y Ft(\001\000)832 2772 y Fx(m)818 2843 y FJ(1)p Fx(;)p FJ(1)908 2738 y Ft(\001)970 2812 y FK(+)1061 2738 y Ft(\000)1126 2772 y Fx(n)1102 2843 y FJ(1)p Fx(;)p FJ(1)1192 2738 y Ft(\001\000)1276 2772 y Fx(m)1289 2843 y FJ(4)1338 2738 y Ft(\001)1380 2711 y(\021)1434 2738 y(\000)1476 2812 y FD(q)1520 2775 y FJ(3)1579 2812 y FC(\000)g FK(4)p FD(q)1759 2775 y FJ(2)1819 2812 y FK(+)g(6)p FD(q)k FC(\000)c FK(4)2156 2738 y Ft(\001)380 3011 y FK(+)515 2975 y FJ(4)p 480 2990 V 480 3042 a Fx(nm)596 2910 y Ft(\020)650 2937 y(\000)715 2971 y Fx(n)692 3042 y FJ(2)p Fx(;)p FJ(1)782 2937 y Ft(\001)o(\000)865 2971 y Fx(m)879 3042 y FJ(2)927 2937 y Ft(\001)989 3011 y FK(+)1080 2937 y Ft(\000)1122 2971 y Fx(n)1126 3042 y FJ(2)1165 2937 y Ft(\001)o(\000)1262 2971 y Fx(m)1248 3042 y FJ(2)p Fx(;)p FJ(1)1338 2937 y Ft(\001)1380 2910 y(\021)1434 2937 y(\000)1476 3011 y FD(q)1520 2973 y FJ(3)1579 3011 y FC(\000)g FK(3)p FD(q)1759 2973 y FJ(2)1819 3011 y FK(+)g(5)p FD(q)k FC(\000)c FK(4)2156 2937 y Ft(\001)380 3186 y FK(+)515 3150 y FJ(4)p 480 3165 V 480 3217 a Fx(nm)596 3112 y Ft(\000)661 3145 y Fx(n)637 3217 y FJ(2)p Fx(;)p FJ(1)727 3112 y Ft(\001\000)825 3145 y Fx(m)811 3217 y FJ(2)p Fx(;)p FJ(1)901 3112 y Ft(\001\000)984 3186 y FD(q)1028 3148 y FJ(5)1088 3186 y FK(+)g(2)p FD(q)1268 3148 y FJ(4)1328 3186 y FC(\000)f FK(10)p FD(q)1552 3148 y FJ(3)1613 3186 y FK(+)h(7)p FD(q)1793 3148 y FJ(2)1853 3186 y FK(+)g(11)p FD(q)k FC(\000)c FK(17)2280 3112 y Ft(\001)2322 3186 y FD(;)125 3345 y Fp(wher)-5 b(e)37 b FD(q)f FK(=)c([)p FD(k)s FK(])i FC(2)e FD(K)7 b FK(\()p FC(V)g FD(ar)1063 3360 y Fx(k)1106 3345 y FK(\))37 b Fp(is)f(the)h(motive)h(of)f(the)g(a\016ne)g(line)f (and)i(the)f(multinomial)i(numb)-5 b(ers)125 3453 y(ar)g(e)33 b(de\014ne)-5 b(d)34 b(in)e(Cor)-5 b(ol)5 b(lary)1101 3471 y SDict begin H.S end 1101 3471 a 0 TeXcolorgray -18 x Fp(6.7)p 0 TeXcolorgray 1222 3394 a SDict begin H.R end 1222 3394 a 1222 3453 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6.7) cvn H.B /ANN pdfmark end 1222 3453 a Fp(.)224 3651 y FK(In)36 b(Section)669 3651 y SDict begin H.S end 669 3651 a 0 TeXcolorgray FK(9)p 0 TeXcolorgray 715 3592 a SDict begin H.R end 715 3592 a 715 3651 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.9) cvn H.B /ANN pdfmark end 715 3651 a FK(,)j(w)m(e)e(mo)m(v)m(e)i(from)d(the)h(motiv)m(e)h(of)f (the)g(mo)s(duli)d(space)k(of)f(irreducible)d(represen-)125 3759 y(tations,)e FB(M)550 3726 y FJ(irr)630 3759 y FK(\(\000)p FD(;)15 b FK(SL)869 3773 y FJ(4)909 3759 y FK(\()p FD(k)s FK(\)\),)34 b(to)f(the)f(motiv)m(e)h(of)f(the)g(total)h(c)m(haracter)g (v)-5 b(ariet)m(y)33 b FB(M)p FK(\(\000)p FD(;)15 b FK(SL)3212 3773 y FJ(4)3252 3759 y FK(\()p FD(k)s FK(\)\).)47 b(F)-8 b(or)125 3867 y(that)33 b(purp)s(ose,)g(w)m(e)g(use)g(the)g(w)m (ell-kno)m(wn)f(fact)i(that)g FB(M)p FK(\(\000)p FD(;)15 b FK(SL)2342 3881 y FJ(4)2382 3867 y FK(\()p FD(k)s FK(\)\))34 b(parametrizes)f(isomorphism)125 3975 y(classes)g(of)h(semi-simple)d (represen)m(tations)j(and,)g(therefore,)h(the)f(calculation)f(can)h(b)s (e)f(reduced)g(to)125 4082 y(com)m(binations)27 b(of)i(the)g(previous)e (results)h(for)g(the)h(c)m(haracter)h(v)-5 b(arieties)28 b(of)h(irreducible)d(represen)m(ta-)125 4190 y(tions)j(for)h(all)g(the) g(ranks)g(1)c FC(\024)f FD(r)i FC(\024)e FK(4.)224 4334 y(W)-8 b(e)40 b(end)d(this)g(section)i(with)d(some)j(\014nal)e(w)m (ords)g(ab)s(out)h(the)h(higher)e(rank)g(case)j FD(r)g(>)e FK(4.)65 b(The)125 4442 y(results)35 b(pro)m(v)m(ed)h(in)f(this)g(pap)s (er)g(ab)s(out)h(the)h(structure)e(of)i(the)f(irreducible)d(represen)m (tations)j(and)125 4550 y(their)g(strati\014cation)i(according)g(to)g (the)h(t)m(yp)s(e)f(of)g(their)f(semi-simple)e(\014ltration)h(are)j (completely)125 4658 y(general)f(and)f(ma)m(y)i(b)s(e)f(used)f(in)g (higher)g(rank.)64 b(Ho)m(w)m(ev)m(er,)42 b(t)m(w)m(o)e(issues)d(prev)m (en)m(t)i(the)f(metho)s(d)g(to)125 4765 y(directly)21 b(generalize)i(to)h(these)f(cases.)39 b(The)23 b(\014rst)f(one)h(is)f (purely)f(computational,)j(since)f(the)g(n)m(um)m(b)s(er)125 4873 y(of)30 b(t)m(yp)s(es)h(to)h(b)s(e)e(analyzed)h(gro)m(ws)g(exp)s (onen)m(tially)e(with)h(rank,)g(so)h(to)h(consider)d(higher)h(rank)g (there)125 4981 y(is)25 b(an)h(extrinsic)e(limitation)g(imp)s(osed)h(b) m(y)h(the)g(a)m(v)-5 b(ailable)25 b(computational)h(p)s(o)m(w)m(er.)39 b(The)26 b(second)g(one)125 5089 y(is)32 b(m)m(uc)m(h)i(subtler)f(and)g (is)g(hidden)e(in)i(Section)1781 5089 y SDict begin H.S end 1781 5089 a 0 TeXcolorgray FK(4)p 0 TeXcolorgray 1826 5031 a SDict begin H.R end 1826 5031 a 1826 5089 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.4) cvn H.B /ANN pdfmark end 1826 5089 a 34 w FK(and)g(Prop)s(osition)2533 5089 y SDict begin H.S end 2533 5089 a 0 TeXcolorgray FK(4.5)p 0 TeXcolorgray 2649 5031 a SDict begin H.R end 2649 5031 a 2649 5089 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.5) cvn H.B /ANN pdfmark end 2649 5089 a FK(.)52 b(The)33 b(k)m(ey)i(p)s(oin)m(t)e(is)g(that,)125 5197 y(more)24 b(precisely)-8 b(,)25 b(in)f(general)g(the)h(\014bration)e (\()1688 5197 y SDict begin H.S end 1688 5197 a 0 TeXcolorgray FK(2)p 0 TeXcolorgray 1734 5139 a SDict begin H.R end 1734 5139 a 1734 5197 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.2) cvn H.B /ANN pdfmark end 1734 5197 a FK(\))i(is)e(not)i(lo)s (cally)e(trivial)g(in)g(the)i(Zariski)e(top)s(ology)-8 b(.)39 b(In)125 5305 y(the)32 b(general)h(case,)h(this)d(\014bration)g (presen)m(ts)i(non-trivial)d(mono)s(drom)m(y)i(coming)g(from)g(the)h (action)125 5413 y(on)j FB(M)352 5427 y Fx(\034)432 5413 y FK(b)m(y)g(p)s(erm)m(uting)e(isomorphic)h(irreducible)e(represen)m (tations.)58 b(In)36 b(ranks)g FD(r)h FC(\024)e FK(4,)k(the)d(only)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)560 b(5)p 0 TeXcolorgray 125 266 a FK(rep)s(eated)27 b(irreducible)c(represen)m(tations)28 b(that)f(ma)m(y)h(o)s(ccur)f(in)f FB(M)2394 280 y Fx(\034)2464 266 y FK(are)i(1-dimensional)d(\(w)m(e)j(get)g(rid)125 374 y(of)c(the)g(case)h(of)f(t)m(w)m(o)i(2-dimensional)21 b(represen)m(tations)j(in)f(Prop)s(osition)2597 374 y SDict begin H.S end 2597 374 a 0 TeXcolorgray FK(8.1)p 0 TeXcolorgray 2713 315 a SDict begin H.R end 2713 315 a 2713 374 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.8.1) cvn H.B /ANN pdfmark end 2713 374 a FK(\))i(and,)g(since)e(the)h(mo) s(duli)125 482 y(space)30 b(of)h(suc)m(h)f(represen)m(tations)g(is)g (just)g(a)g(p)s(oin)m(t,)g(the)h(action)f(b)m(y)h(p)s(erm)m(utations)e (is)g(trivial.)224 625 y(Despite)j(these)g(di\016culties,)e(the)i (approac)m(h)g(dev)m(elop)s(ed)f(in)f(this)g(pap)s(er)h(can)h(b)s(e)f (used)g(to)h(tac)m(kle)125 733 y(the)43 b(general)g(rank)g(case.)79 b(A)44 b(more)f(in)m(v)m(olv)m(ed)g(analysis)e(of)j(this)e(p)s(erm)m (utation)g(action)h(ma)m(y)h(b)s(e)125 841 y(p)s(erformed)28 b(in)h(order)h(to)h(understand)d(the)i(mono)s(drom)m(y)g(action,)h(and) e(from)h(it)g(the)g(motiv)m(e)h(of)f(the)125 949 y(total)39 b(space)g(ma)m(y)g(b)s(e)e(computed)h(as)h(in)e([)1623 949 y SDict begin H.S end 1623 949 a 0 TeXcolorgray FK(22)p 0 TeXcolorgray 1714 890 a SDict begin H.R end 1714 890 a 1714 949 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.lomune) cvn H.B /ANN pdfmark end 1714 949 a FK(].)65 b(T)-8 b(o)39 b(this)e(extend,)k(our)d(approac)m(h)g(is)g(suitable)f (for)125 1057 y(studying)28 b(the)j(higher)e(rank)h(case.)224 1200 y(The)h(results)e(of)i(this)f(pap)s(er)g(also)g(hold)g(for)h(an)f (algebraically)g(closed)h(\014eld)e(of)i(p)s(ositiv)m(e)f(c)m(harac-) 125 1308 y(teristic)h FD(p)d(>)f FK(0,)33 b(as)g(long)e(as)h FD(n;)15 b(m;)g(r)35 b FK(are)e(coprime)e(with)g FD(p)p FK(.)45 b(The)31 b(relev)-5 b(an)m(t)33 b(p)s(oin)m(t)e(is)g(that)h(w)m (e)h(tak)m(e)125 1416 y(ro)s(ots)d(of)h(unit)m(y)e(and)h(w)m(e)h(coun)m (t)g(them.)125 1559 y FL(Ac)m(kno)m(wledgemen)m(ts.)39 b FK(The)27 b(authors)h(w)m(an)m(t)g(to)h(thank)e(Eduardo)g(F)-8 b(ern\023)-45 b(andez-F)-8 b(uertes)29 b(and)e(Ma-)125 1667 y(rina)21 b(Logares)j(for)f(v)m(ery)g(useful)e(con)m(v)m (ersations)j(around)d(the)j(metho)s(ds)e(of)h(this)e(pap)s(er.)38 b(W)-8 b(e)24 b(sp)s(ecially)125 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FD(;)f FK(GL)3338 280 y Fx(r)3376 266 y FK(\),)29 b(and)125 374 y(the)h(sets)527 521 y FD(R)596 535 y Fx(r)659 521 y FK(=)25 b FD(R)825 483 y FJ(\()p Fx(\030)883 492 y FG(1)918 483 y Fx(;\030)969 492 y FG(2)1003 483 y FJ(\))824 543 y Fx(r)1060 521 y FK(=)g FC(f)p FK(\()p FD(A;)15 b(B)5 b FK(\))26 b FC(2)f FD(R)q FK(\()p Fz(Z)1735 535 y Fx(n)1798 521 y FD(?)20 b Fz(Z)1928 535 y Fx(m)1991 521 y FD(;)15 b FK(GL)2159 535 y Fx(r)2197 521 y FK(\))p FC(j)g FK(det)h FD(A)26 b FK(=)f FD(\030)2644 535 y FJ(1)2683 521 y FD(;)15 b FK(det)h FD(B)30 b FK(=)24 b FD(\030)3099 535 y FJ(2)3139 521 y FC(g)p FD(;)125 668 y FK(for)g FD(\030)298 682 y FJ(1)337 668 y FD(;)15 b(\030)417 682 y FJ(2)482 668 y FC(2)25 b FD(k)618 635 y Fs(\003)683 668 y FK(\(w)m(e)g(usually)e(drop)h FD(\030)1399 682 y FJ(1)1438 668 y FD(;)15 b(\030)1518 682 y FJ(2)1583 668 y FK(from)24 b(the)h(notation\).)40 b(If)24 b(w)m(e)i(c)m(hange)g FD(\030)2925 682 y FJ(1)2964 668 y FD(;)15 b(\030)3044 682 y FJ(2)3108 668 y FK(to)26 b FD(\030)3258 635 y Fs(0)3254 693 y FJ(1)3319 668 y FK(=)f FD(\025)3468 682 y FJ(1)3507 668 y FD(\030)3547 682 y FJ(1)3586 668 y FK(,)125 799 y FD(\030)169 766 y Fs(0)165 823 y FJ(2)229 799 y FK(=)g FD(\025)378 813 y FJ(2)417 799 y FD(\030)457 813 y FJ(2)497 799 y FK(,)30 b(where)g FD(\025)868 751 y FJ(1)p Fx(=r)868 825 y FJ(1)1002 799 y FC(2)25 b FD(\026)1143 813 y Fx(n)1189 799 y FK(,)31 b FD(\025)1298 751 y FJ(1)p Fx(=r)1298 825 y FJ(2)1432 799 y FC(2)25 b FD(\026)1573 813 y Fx(m)1639 799 y FK(,)31 b(the)f(v)-5 b(ariet)m(y)31 b FD(R)2221 813 y Fx(r)2289 799 y FK(remains)e(the)i(same)g(\(isomorphic\).)125 962 y SDict begin H.S end 125 962 a 125 962 a SDict begin 13 H.A end 125 962 a 125 962 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.2) cvn /DEST pdfmark end 125 962 a 0 TeXcolorgray FL(Prop)s(osition)36 b(3.2.)p 0 TeXcolorgray 42 w Fp(Ther)-5 b(e)33 b(is)g(an)g(isomorphism)1360 1114 y FC(R)1437 1077 y FJ(irr)1437 1137 y Fx(r)1516 1114 y FK(\()p FD($)s FK(\))1690 1089 y FC(\030)1690 1119 y FK(=)1786 1114 y FD(R)1856 1077 y FJ(irr)1855 1137 y Fx(r)1960 1114 y FC(\032)25 b FD(R)2126 1077 y FJ(\()p Fx(\030)2184 1086 y FG(1)2218 1077 y Fx(;\030)2269 1086 y FG(2)2304 1077 y FJ(\))2125 1137 y Fx(r)2350 1114 y FD(;)125 1268 y Fp(for)33 b FD(\030)310 1282 y FJ(1)374 1268 y FK(=)25 b FD($)548 1235 y Fs(\000)p Fx(r)r(=n)719 1268 y Fp(,)32 b FD(\030)819 1282 y FJ(2)884 1268 y FK(=)25 b FD($)1058 1235 y Fs(\000)p Fx(r)r(=m)1248 1268 y Fp(.)p 0 TeXcolorgray 125 1463 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(Since)30 b FD(A)711 1430 y Fx(n)783 1463 y FK(=)25 b FD(B)953 1430 y Fx(m)1044 1463 y FK(=)g FD($)18 b FK(Id)o(,)31 b(if)e(w)m(e)h(\014x)g FD(n)p FK(-th)g(and)f FD(m)p FK(-th)h(ro)s(ots)h (of)f FD($)2732 1430 y Fs(\000)p FJ(1)2826 1463 y FK(,)g(denoted)g(b)m (y)g FD(\021)3395 1477 y FJ(1)3465 1463 y FK(and)125 1571 y FD(\021)170 1585 y FJ(2)244 1571 y FK(resp)s(ectiv)m(ely)-8 b(,)37 b(then)e(w)m(e)g(ha)m(v)m(e)i(that)f(\()p FD(\021)1611 1585 y FJ(1)1650 1571 y FD(A)p FK(\))1753 1538 y Fx(n)1834 1571 y FK(=)d(\()p FD(\021)2018 1585 y FJ(2)2058 1571 y FD(B)5 b FK(\))2167 1538 y Fx(m)2267 1571 y FK(=)33 b(Id)o(.)56 b(Hence,)37 b(the)f(pair)d(\()p FD(\021)3269 1585 y FJ(1)3309 1571 y FD(A;)15 b(\021)3462 1585 y FJ(2)3503 1571 y FD(B)5 b FK(\))125 1679 y(de\014nes)29 b(a)h(represen)m(tation)g (of)g FD(R)q FK(\()p Fz(Z)1365 1693 y Fx(n)1427 1679 y FD(?)19 b Fz(Z)1556 1693 y Fx(m)1619 1679 y FD(;)c FK(GL)1787 1693 y Fx(r)1825 1679 y FK(\).)41 b(No)m(w,)31 b(observ)m(e)f(that)h(det\()p FD(\021)2884 1693 y FJ(1)2924 1679 y FD(A)p FK(\))26 b(=)f FD(\021)3197 1646 y Fx(r)3194 1703 y FJ(1)3250 1679 y FK(det)q(\()p FD(A)p FK(\))h(=)125 1809 y FD(\021)173 1776 y Fx(r)170 1834 y FJ(1)251 1809 y FK(=)40 b FD(\030)402 1823 y FJ(1)441 1809 y FK(,)i(and)d(similarly)d (det\()p FD(\021)1281 1823 y FJ(2)1321 1809 y FD(B)5 b FK(\))40 b(=)g FD(\030)1621 1823 y FJ(2)1660 1809 y FK(,)i(so)e(the)f(pair)f(\()p FD(\021)2289 1823 y FJ(1)2329 1809 y FD(A;)15 b(\021)2482 1823 y FJ(2)2522 1809 y FD(B)5 b FK(\))40 b(actually)f(lies)f(in)g FD(R)3377 1761 y FJ(\()p Fx(\030)3435 1770 y FG(1)3469 1761 y Fx(;\030)3520 1770 y FG(2)3555 1761 y FJ(\))3376 1821 y Fx(r)3586 1809 y FK(.)125 1917 y(Clearly)-8 b(,)32 b(the)i(image)f(of)g(this)f (assignmen)m(t)h(\()p FD(A;)15 b(B)5 b FK(\))30 b FC(7!)g FK(\()p FD(\021)2126 1931 y FJ(1)2166 1917 y FD(A;)15 b(\021)2319 1931 y FJ(2)2359 1917 y FD(B)5 b FK(\))33 b(coincides)f(with)g(the)h(subset)f(of)125 2025 y(irreducible)27 b(represen)m(tations.)3541 2169 y Fo(\003)125 2331 y SDict begin H.S end 125 2331 a 125 2331 a SDict begin 13 H.A end 125 2331 a 125 2331 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.3) cvn /DEST pdfmark end 125 2331 a 0 TeXcolorgray FL(Corollary)36 b(3.3.)p 0 TeXcolorgray 42 w Fp(If)d FD(\032)25 b FK(=)g(\()p FD(A;)15 b(B)5 b FK(\))26 b FC(2)f(R)1500 2298 y FJ(irr)1500 2354 y Fx(r)1579 2331 y Fp(,)32 b(then)h FD(A)g Fp(and)g FD(B)k Fp(ar)-5 b(e)34 b(diagonalizable.)p 0 TeXcolorgray 125 2527 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(An)m(y)35 b(represen)m(tation)f(of)g(a)g(\014nite)g(group)f(is)g(completely)h (reducible)e(and,)i(if)g(the)g(group)f(is)125 2635 y(ab)s(elian,)28 b(the)j(irreducible)c(represen)m(tations)j(are)h(1-dimensional.)1114 b Fo(\003)125 2798 y SDict begin H.S end 125 2798 a 125 2798 a SDict begin 13 H.A end 125 2798 a 125 2798 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.4) cvn /DEST pdfmark end 125 2798 a 0 TeXcolorgray FL(Remark)25 b(3.4.)p 0 TeXcolorgray 36 w FD(A)e FK(and)f FD(B)27 b FK(are)c(diagonalizable,) g(but)f(they)h(cannot)g(b)s(e)f(diagonalizable)g(in)f(the)i(same)125 2905 y(basis.)38 b(Indeed,)26 b(they)h(cannot)g(share)g(ev)m(en)g(an)f (eigen)m(v)m(ector)j(since,)e(otherwise,)g(the)f(represen)m(tation)125 3013 y FD(\032)k FK(w)m(ould)f(b)s(e)h(reducible.)224 3212 y(By)e(Prop)s(osition)850 3212 y SDict begin H.S end 850 3212 a 0 TeXcolorgray FK(3.2)p 0 TeXcolorgray 966 3153 a SDict begin H.R end 966 3153 a 966 3212 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.2) cvn H.B /ANN pdfmark end 966 3212 a FK(,)g(w)m(e)g(fo)s(cus)e(on)h FD(R)1571 3226 y Fx(r)1609 3212 y FK(.)40 b(Observ)m(e)27 b(that)g(the)h(p)s (ossible)d(eigen)m(v)-5 b(alues)26 b(of)i(\()p FD(A;)15 b(B)5 b FK(\))26 b FC(2)125 3320 y FD(R)194 3334 y Fx(r)267 3320 y FK(form)35 b(a)h(discrete)g(set,)h(since)e FD(A)1377 3287 y Fx(n)1458 3320 y FK(=)f(Id)h(implies)e(that)j(the)g(eigen)m(v)-5 b(alues)35 b(of)h FD(A)g FK(are)g FD(n)p FK(-th)f(ro)s(ots)125 3428 y(of)g(unit)m(y)-8 b(,)35 b(and)f(similarly)e(for)i FD(B)5 b FK(.)54 b(This)33 b(allo)m(ws)h(a)h(re\014nemen)m(t)g(of)g (the)g(strati\014cation)g(of)g(\()3319 3428 y SDict begin H.S end 3319 3428 a 0 TeXcolorgray FK(4)p 0 TeXcolorgray 3364 3369 a SDict begin H.R end 3364 3369 a 3364 3428 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end 3364 3428 a FK(\).)55 b(Let)125 3554 y FD(\032)37 b FK(=)f(\()p FD(A;)15 b(B)5 b FK(\))38 b FC(2)f FD(R)773 3568 y Fx(r)848 3554 y FK(and)g(let)g Fn(\017)g FK(=)1359 3480 y Ft(\010)1412 3554 y FD(\017)1449 3513 y Fx(a)1486 3522 y FG(1)1449 3580 y FJ(1)1525 3554 y FD(;)15 b(:)g(:)g(:)i(;)e(\017)1764 3505 y Fx(a)1801 3513 y Fq(p)1764 3566 y Fx(p)1842 3480 y Ft(\011)1932 3554 y FK(and)37 b Fn(")g FK(=)2309 3480 y Ft(\010)2362 3554 y FD(")2404 3513 y Fx(b)2434 3522 y FG(1)2404 3580 y FJ(1)2474 3554 y FD(;)15 b(:)g(:)g(:)h(;)f(")2717 3505 y Fx(b)2747 3513 y Fq(q)2717 3566 y Fx(q)2787 3480 y Ft(\011)2877 3554 y FK(b)s(e)37 b(the)h(eigen)m(v)-5 b(alues)125 3662 y(of)41 b FD(A)g FK(and)g FD(B)46 b FK(resp)s(ectiv)m(ely)40 b(\(as)i(m)m(ultisets)e(with)g(rep)s (etitions\),)j(that)f(will)d(b)s(e)i(collected)g(in)f(the)125 3770 y Fp(c)-5 b(on\014gur)g(ation)34 b(of)f(eigenvalues)125 3918 y FK(\(8\))1667 3918 y SDict begin H.S end 1667 3918 a 1667 3918 a SDict begin 13 H.A end 1667 3918 a 1667 3918 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.8) cvn /DEST pdfmark end 1667 3918 a FD(\024)26 b FK(=)f(\()p Fn(\017)p FD(;)15 b Fn(")p FK(\))p FD(:)125 4065 y FK(This)28 b(implies)g(that)j FD(R)907 4079 y Fx(r)975 4065 y FK(can)g(b)s(e)e (decomp)s(osed)h(as)h(a)g(\014nite)e(disjoin)m(t)g(union)125 4226 y(\(9\))1111 4226 y SDict begin H.S end 1111 4226 a 1111 4226 a SDict begin 13 H.A end 1111 4226 a 1111 4226 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.9) cvn /DEST pdfmark end 1111 4226 a FD(R)1180 4240 y Fx(r)1243 4226 y FK(=)1339 4140 y Ft(G)1369 4330 y Fx(\024)1455 4226 y FD(R)1524 4240 y Fx(\024)1569 4226 y FD(;)107 b FK(where)60 b FD(R)2063 4240 y Fx(\024)2133 4226 y FK(=)2261 4140 y Ft(G)2229 4336 y Fx(\034)8 b Fs(2T)2354 4344 y Fq(\024)2409 4226 y FD(R)q FK(\()p FD(\034)i FK(\))p FD(:)125 4470 y FK(Here)34 b FD(R)q FK(\()p FD(\034)10 b FK(\))31 b FC(\032)f FD(R)734 4484 y Fx(r)805 4470 y FK(is,)k(as)g(in)e(\()1185 4470 y SDict begin H.S end 1185 4470 a 0 TeXcolorgray FK(4)p 0 TeXcolorgray 1231 4411 a SDict begin H.R end 1231 4411 a 1231 4470 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end 1231 4470 a FK(\),)j(the)e(collection)g(of)h(represen)m(tations)g(of) f FD(R)2808 4484 y Fx(r)2880 4470 y FK(of)g(t)m(yp)s(e)h FD(\034)41 b FK(=)30 b(\()p FD(\030)t(;)15 b(\033)s FK(\),)125 4578 y(and)29 b FC(T)351 4592 y Fx(\024)426 4578 y FK(is)g(the)i (collection)f(of)g(t)m(yp)s(es)h(whose)f(con\014guration)g(of)g(eigen)m (v)-5 b(alues)31 b FD(\033)i FK(is)c(tak)m(en)j(from)e FD(\024)p FK(.)224 4721 y(F)-8 b(rom)34 b(this)f(decomp)s(osition,)g(w) m(e)h(can)g(understand)d(the)j(virtual)e(class)h(of)h FC(R)2919 4688 y FJ(irr)2919 4744 y Fx(r)2998 4721 y FK(\()p FD($)s FK(\))d FC(\032)f FD(R)3347 4735 y Fx(r)3385 4721 y FK(.)50 b(Ob-)125 4829 y(serv)m(e)33 b(that,)g(if)f FC(T)737 4796 y Fs(\003)715 4851 y Fx(\024)806 4829 y FC(\032)c(T)955 4843 y Fx(\024)1032 4829 y FK(are)k(the)h(t)m(yp)s(es)f (whose)h(shap)s(e)e(do)s(es)h(not)h(corresp)s(ond)e(to)i(an)g (irreducible)125 4939 y(represen)m(tation,)41 b(then)d FD(R)1040 4906 y FJ(red)1039 4961 y Fx(\024)1181 4939 y FK(=)1342 4871 y Ft(F)1291 5022 y Fx(\034)8 b Fs(2T)1433 5003 y Fr(\003)1416 5039 y Fq(\024)1484 4939 y FD(R)q FK(\()p FD(\034)i FK(\))39 b(is)f(the)h(collection)f(of)h(reducible)e (represen)m(tations)h(of)125 5135 y FD(R)194 5149 y Fx(\024)238 5135 y FK(,)31 b(and)f(the)g(irreducible)d(represen)m(tations)k(are)f FD(R)1929 5102 y FJ(irr)1928 5158 y Fx(\024)2033 5135 y FK(=)25 b FD(R)2198 5149 y Fx(\024)2263 5135 y FC(\000)20 b FD(R)2424 5102 y FJ(red)2423 5158 y Fx(\024)2526 5135 y FK(.)41 b(Hence,)31 b(w)m(e)g(ha)m(v)m(e)g(that)125 5310 y(\(10\))527 5310 y SDict begin H.S end 527 5310 a 527 5310 a SDict begin 13 H.A end 527 5310 a 527 5310 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.10) cvn /DEST pdfmark end 527 5310 a FC(R)604 5272 y FJ(irr)604 5332 y Fx(r)682 5310 y FK(\()p FD($)s FK(\))26 b(=)f FD(R)1022 5272 y FJ(irr)1021 5332 y Fx(r)1126 5310 y FK(=)1222 5224 y Ft(G)1252 5415 y Fx(\024)1338 5310 y FD(R)1408 5272 y FJ(irr)1407 5332 y Fx(\024)1512 5310 y FK(=)1608 5224 y Ft(G)1638 5415 y Fx(\024)1724 5209 y Ft(\020)1778 5310 y FD(R)1847 5324 y Fx(\024)1912 5310 y FC(\000)20 b FD(R)2073 5272 y FJ(red)2072 5332 y Fx(\024)2175 5209 y Ft(\021)2255 5310 y FK(=)2351 5224 y Ft(G)2381 5415 y Fx(\024)2467 5209 y Ft(\020)2521 5310 y FD(R)2590 5324 y Fx(\024)2655 5310 y FC(\000)2785 5224 y Ft(G)2746 5421 y Fx(\034)8 b Fs(2T)2888 5402 y Fr(\003)2871 5437 y Fq(\024)2939 5310 y FD(R)q FK(\()p FD(\034)i FK(\))3129 5209 y Ft(\021)3184 5310 y FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(10)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 224 266 a FK(The)30 b(virtual)f(class)h([)p FD(R)1014 280 y Fx(\024)1059 266 y FK(])h(is)e(v)m(ery)i(easy)g(to)g(compute.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)h(that)1292 423 y FD(R)1361 437 y Fx(\024)1431 423 y FK(=)25 b(\()q(GL)1691 437 y Fx(r)1744 423 y FC(\001)p FK(\006)1835 437 y Fm(\017)1873 423 y FK(\))20 b FC(\002)g FK(\()q(GL)2183 437 y Fx(r)2236 423 y FC(\001)p FK(\006)2327 437 y Fm(")2368 423 y FK(\))c FD(;)125 580 y FK(where)35 b(\006)459 594 y Fm(\017)533 580 y FK(is)h(the)g(diagonal)g(matrix)f(with)g(eigen)m(v)-5 b(alues)37 b Fn(\017)f FK(and)f(GL)2545 594 y Fx(r)2598 580 y FC(\001)p FK(\006)2689 594 y Fm(\017)2763 580 y FK(is)g(its)h(orbit)f(under)g(the)125 688 y(GL)253 702 y Fx(r)291 688 y FK(-action)g(b)m(y)f(conjugation,)h(and)f(analogously) g(for)g Fn(")p FK(.)52 b(The)34 b(stabilizers)f(of)h(\006)2990 702 y Fm(\017)3062 688 y FK(and)g(\006)3309 702 y Fm(")3384 688 y FK(under)125 796 y(this)23 b(action)i(are)g(GL)837 810 y Fx(a)874 819 y FG(1)928 796 y FC(\002)15 b FD(:)g(:)g(:)10 b FC(\002)f FK(GL)1337 810 y Fx(a)1374 818 y Fq(p)1439 796 y FK(and)24 b(GL)1738 811 y Fx(b)1768 820 y FG(1)1822 796 y FC(\002)15 b FD(:)g(:)g(:)10 b FC(\002)f FK(GL)2230 811 y Fx(b)2260 819 y Fq(q)2299 796 y FK(,)26 b(resp)s(ectiv)m(ely)-8 b(.)38 b(Therefore,)26 b(w)m(e)f(ha)m(v)m(e)125 904 y(that)125 1091 y(\(11\))1013 1091 y SDict begin H.S end 1013 1091 a 1013 1091 a SDict begin 13 H.A end 1013 1091 a 1013 1091 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.11) cvn /DEST pdfmark end 1013 1091 a 15 w FK([)p FD(R)1122 1105 y Fx(\024)1167 1091 y FK(])g(=)1539 1029 y([GL)1692 1043 y Fx(r)1730 1029 y FK(])p 1323 1070 648 4 v 1323 1153 a([GL)1477 1167 y Fx(a)1514 1176 y FG(1)1553 1153 y FK(])15 b FC(\001)g(\001)g(\001)i FK([GL)1868 1167 y Fx(a)1905 1175 y Fq(p)1946 1153 y FK(])2001 1091 y FC(\001)2264 1029 y FK([GL)2417 1043 y Fx(r)2455 1029 y FK(])p 2057 1070 632 4 v 2057 1153 a([GL)2210 1168 y Fx(b)2240 1177 y FG(1)2279 1153 y FK(])e FC(\001)g(\001)g(\001)i FK([GL)2594 1168 y Fx(b)2624 1176 y Fq(q)2663 1153 y FK(])2698 1091 y FD(:)224 1356 y FK(Using)29 b(\()514 1356 y SDict begin H.S end 514 1356 a 0 TeXcolorgray FK(10)p 0 TeXcolorgray 605 1298 a SDict begin H.R end 605 1298 a 605 1356 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 605 1356 a FK(\),)i(w)m(e)f(get)h (that,)f(in)f(order)g(to)h(compute)g(the)g(virtual)e(class)i(of)f FC(R)2865 1323 y FJ(irr)2865 1379 y Fx(r)2944 1356 y FK(\()p FD($)s FK(\),)i(it)e(is)g(enough)125 1464 y(to)37 b(compute)h(the)f(virtual)f(classes)h(of)h FD(R)q FK(\()p FD(\034)10 b FK(\),)39 b(for)e(all)f FD(\024)i FK(and)e FD(\034)47 b FC(2)36 b(T)2554 1431 y Fs(\003)2531 1487 y Fx(\024)2594 1464 y FK(.)61 b(As)37 b(w)m(e)h(will)c(sho)m(w)j(in)f (the)125 1573 y(follo)m(wing)19 b(section,)k(this)c(amoun)m(ts)i(to)g (a)h(com)m(binatorial)d(problem)g(and)h(the)h(kno)m(wledge)g(of)g([)p FC(R)3359 1540 y FJ(irr)3359 1595 y Fx(s)3438 1573 y FK(\()p FD($)s FK(\)])125 1681 y(for)30 b FD(s)25 b(<)f(r)s FK(,)31 b(so)f(the)h(computation)f(can)h(b)s(e)e(p)s(erformed)g (recursiv)m(ely)-8 b(.)224 1824 y(F)g(rom)34 b(this)e(computation,)i (one)g(can)g(also)f(compute)h(the)f(virtual)f(class)h(of)g FB(M)2977 1791 y FJ(irr)2977 1847 y Fx(r)3087 1824 y FK(=)d FD(R)3258 1791 y FJ(irr)3257 1847 y Fx(r)3358 1824 y FA(\014)22 b FK(SL)3548 1838 y Fx(r)3586 1824 y FK(.)125 1932 y(Indeed,)j(the)h(action)f(of)h(PGL)1149 1946 y Fx(r)1212 1932 y FK(=)f(SL)1415 1946 y Fx(r)1468 1932 y FD(=)-5 b(Z)7 b FK(\(SL)1720 1946 y Fx(r)1758 1932 y FK(\))25 b(is)g(free)g(and)g(has)g(closed)g(orbits)f(on)h(the)h (irreducible)125 2040 y(represen)m(tations.)40 b(Hence,)30 b(w)m(orking)d(analogously)h(to)i(Prop)s(osition)c(7.3)k(of)f([)2785 2040 y SDict begin H.S end 2785 2040 a 0 TeXcolorgray FK(8)p 0 TeXcolorgray 2830 1982 a SDict begin H.R end 2830 1982 a 2830 2040 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.GP:2018) cvn H.B /ANN pdfmark end 2830 2040 a FK(])g(for)f(arbitrary) g(rank,)125 2148 y(w)m(e)i(get)i(that)1533 2324 y([)p FB(M)1653 2287 y FJ(irr)1653 2347 y Fx(r)1733 2324 y FK(])25 b(=)1929 2263 y([)p FD(R)2024 2230 y FJ(irr)2023 2285 y Fx(r)2103 2263 y FK(])p 1889 2304 279 4 v 1889 2387 a([PGL)2104 2401 y Fx(r)2142 2387 y FK(])2178 2324 y FD(:)125 2526 y FK(This)j(action)j(resp)s(ects)f(the)h (strati\014cation)f(\()1680 2526 y SDict begin H.S end 1680 2526 a 0 TeXcolorgray FK(9)p 0 TeXcolorgray 1726 2467 a SDict begin H.R end 1726 2467 a 1726 2526 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 1726 2526 a FK(\))g(so)h(de\014ning)d FB(M)2341 2493 y FJ(irr)2341 2548 y Fx(\024)2446 2526 y FK(=)d FD(R)2612 2493 y FJ(irr)2611 2548 y Fx(\024)2710 2526 y FA(\014)20 b FK(SL)2898 2540 y Fx(r)2967 2526 y FK(w)m(e)31 b(ha)m(v)m(e)125 2745 y(\(12\))1221 2745 y SDict begin H.S end 1221 2745 a 1221 2745 a SDict begin 13 H.A end 1221 2745 a 1221 2745 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.12) cvn /DEST pdfmark end 1221 2745 a FK([)p FB(M)1341 2707 y FJ(irr)1341 2767 y Fx(r)1421 2745 y FK(])25 b(=)1567 2659 y Ft(X)1613 2849 y Fx(\024)1699 2745 y FK([)p FB(M)1819 2707 y FJ(irr)1819 2767 y Fx(\024)1898 2745 y FK(])h(=)2044 2659 y Ft(X)2090 2849 y Fx(\024)2241 2683 y FK([)p FD(R)2336 2650 y FJ(irr)2335 2706 y Fx(\024)2414 2683 y FK(])p 2201 2724 279 4 v 2201 2807 a([PGL)2416 2821 y Fx(r)2454 2807 y FK(])2489 2745 y FD(:)125 2996 y SDict begin H.S end 125 2996 a 125 2996 a SDict begin 13 H.A end 125 2996 a 125 2996 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.5) cvn /DEST pdfmark end 125 2996 a 0 TeXcolorgray FL(Remark)40 b(3.5.)p 0 TeXcolorgray 45 w FK(Consider)34 b(a)j(con\014guration)e(of)h(eigen)m(v)-5 b(alues)36 b FD(\024)f FK(=)f(\()p Fn(\017)p FD(;)15 b Fn(")p FK(\))37 b(with)d(m)m(ultiplicities)e FD(a)3583 3010 y Fx(i)125 3104 y FK(and)37 b FD(b)348 3118 y Fx(j)423 3104 y FK(resp)s(ectiv)m (ely)-8 b(.)65 b(If)39 b(there)f(exists)g FD(i;)15 b(j)45 b FK(suc)m(h)39 b(that)g FD(a)2190 3118 y Fx(i)2244 3104 y FK(+)25 b FD(b)2379 3118 y Fx(j)2454 3104 y FD(>)39 b(r)s FK(,)h(then)e FD(R)2958 3071 y FJ(irr)2957 3127 y Fx(\024)3076 3104 y FK(=)g FC(;)p FK(.)66 b(Indeed,)125 3212 y(if)36 b FD(\032)h FK(=)f(\()p FD(A;)15 b(B)5 b FK(\))38 b FC(2)f FD(R)863 3226 y Fx(\024)908 3212 y FK(,)i(then)e FD(A)h FK(has)f(an)g(eigenspace)h FD(V)57 b FK(of)38 b(dimension)d FD(a)2752 3226 y Fx(i)2780 3212 y FK(,)k(and)e FD(B)42 b FK(has)37 b(another)125 3320 y(eigenspace)32 b FD(W)44 b FK(of)31 b(dimension)e FD(b)1275 3334 y Fx(j)1312 3320 y FK(.)44 b(But)31 b(since)g FD(a)1834 3334 y Fx(i)1884 3320 y FK(+)20 b FD(b)2014 3334 y Fx(j)2078 3320 y FD(>)27 b(r)s FK(,)k FD(V)41 b FC(\\)21 b FD(W)40 b FC(6)p FK(=)27 b(0)k(and)g(th)m(us,)h(there)g(exists)125 3428 y(at)f(least)h(a)f(common)h(eigen)m(v)m(ector)h(to)f FD(A)f FK(and)g FD(B)36 b FK(so)31 b(they)g(are)h(reducible.)41 b(Hence)32 b(w)m(e)f(can)h(discard)125 3536 y(suc)m(h)e FD(\024)g FK(in)f(the)i(form)m(ula)e(\()1043 3536 y SDict begin H.S end 1043 3536 a 0 TeXcolorgray FK(10)p 0 TeXcolorgray 1135 3477 a SDict begin H.R end 1135 3477 a 1135 3536 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 1135 3536 a FK(\).)125 3667 y SDict begin H.S end 125 3667 a 125 3667 a SDict begin 13 H.A end 125 3667 a 125 3667 a SDict begin [/View [/XYZ H.V]/Dest (section.4) cvn /DEST pdfmark end 125 3667 a 1099 3787 a FK(4.)46 b FE(Represent)-6 b(a)g(tions)31 b(of)j(fixed)g(type)224 3985 y FK(Throughout)22 b(this)g(section,)j(w)m (e)f(will)c(\014x)j(a)h(con\014guration)e(of)h(eigen)m(v)-5 b(alues)23 b FD(\024)j FK(=)f(\()p Fn(\017)p FD(;)15 b Fn(")p FK(\))24 b(and)f(a)g(t)m(yp)s(e)125 4093 y FD(\034)43 b FC(2)33 b(T)352 4107 y Fx(\024)396 4093 y FK(,)k(and)d(w)m(e)i(will)d (study)h(the)h(collection)g(of)h Fz(Z)1954 4107 y Fx(n)2020 4093 y FD(?)24 b Fz(Z)2154 4107 y Fx(m)2216 4093 y FK(-represen)m (tations)35 b FD(R)q FK(\()p FD(\034)10 b FK(\).)56 b(Compatible)125 4200 y(with)22 b(the)j(description)d(of)i(the)h(torus)f(knot)g(\()1648 4200 y SDict begin H.S end 1648 4200 a 0 TeXcolorgray FK(5)p 0 TeXcolorgray 1694 4142 a SDict begin H.R end 1694 4142 a 1694 4200 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.5) cvn H.B /ANN pdfmark end 1694 4200 a FK(\),)i(w)m(e)e(will)e (denote)j(b)m(y)f FD(x;)15 b(y)28 b FC(2)d Fz(Z)2796 4214 y Fx(n)2846 4200 y FD(?)8 b Fz(Z)2964 4214 y Fx(m)3051 4200 y FK(the)24 b(generators)125 4308 y(of)30 b(the)h Fz(Z)450 4322 y Fx(n)523 4308 y FK(and)e Fz(Z)764 4322 y Fx(m)857 4308 y FK(parts,)h(resp)s(ectiv)m(ely)-8 b(.)224 4452 y(F)g(ollo)m(wing)30 b(the)h(notation)g(of)g(\()1292 4452 y SDict begin H.S end 1292 4452 a 0 TeXcolorgray FK(3)p 0 TeXcolorgray 1338 4393 a SDict begin H.R end 1338 4393 a 1338 4452 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.3) cvn H.B /ANN pdfmark end 1338 4452 a FK(\),)g(let)f FD(m)1640 4466 y Fx(i;j)1751 4452 y FK(b)s(e)g(the)h(m)m(ultiplicit)m (y)c(of)k(the)g(isot)m(ypic)f(piece)h FD(W)3428 4466 y Fx(i;j)3538 4452 y FK(of)125 4560 y(the)k(semi-simple)e(\014ltration) h(of)h(the)h(t)m(yp)s(e)f FD(\034)10 b FK(.)56 b(The)35 b(corresp)s(onding)e(eigen)m(v)-5 b(alues)35 b(of)h(these)g(pieces)125 4668 y(are)30 b(denoted)h FD(\024)672 4682 y Fx(i;j)778 4668 y FK(=)24 b(\()p FD(\033)960 4682 y Fx(i;j)1041 4668 y FK(\()p FD(x)p FK(\))p FD(;)15 b(\033)1255 4682 y Fx(i;j)1336 4668 y FK(\()p FD(y)s FK(\)\).)42 b(Then)29 b(w)m(e)i(tak)m(e)125 4905 y(\(13\))1251 4905 y SDict begin H.S end 1251 4905 a 1251 4905 a SDict begin 13 H.A end 1251 4905 a 1251 4905 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.13) cvn /DEST pdfmark end 1251 4905 a FC(I)7 b FK(\()p FD(\034)j FK(\))26 b(=)1591 4791 y Fx(s)1550 4819 y Ft(Y)1551 5014 y Fx(i)p FJ(=1)1713 4789 y Fx(s)1746 4799 y Fq(i)1684 4819 y Ft(Y)1681 5014 y Fx(j)t FJ(=1)1819 4905 y FK(Sym)1993 4867 y Fx(m)2055 4877 y Fq(i;j)2148 4804 y Ft(\020)2202 4905 y FD(R)2272 4867 y FJ(irr)2271 4927 y Fx(\024)2312 4937 y Fq(i;j)2390 4804 y Ft(\021)2459 4905 y FD(:)224 5197 y FK(W)-8 b(e)35 b(ha)m(v)m(e)h(a)e(map)g(Gr)990 5211 y Fs(\017)1060 5197 y FK(:)e FD(R)q FK(\()p FD(\034)10 b FK(\))32 b FC(!)f(I)7 b FK(\()p FD(\034)j FK(\))34 b(giv)m(en)g(b)m(y)g FD(\032)d FC(7!)g FK(\()q(Gr)2386 5211 y Fx(i)2414 5197 y FK(\()p FD(\032)p FK(\)\))2567 5224 y Fx(i)2595 5197 y FK(,)k(where)f(the)g (isot)m(ypic)g(com-)125 5305 y(p)s(onen)m(ts)27 b(of)i(the)f(graded)h (represen)m(tations)f(are)h(group)s(ed)e(together)j(in)m(to)e(the)h (symmetric)e(pro)s(duct.)125 5413 y(W)-8 b(e)31 b(will)d(call)h(this)h (map)g(the)g Fp(semi-simpli\014c)-5 b(ation)p FK(.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.11) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(11)p 0 TeXcolorgray 224 266 a FK(The)28 b(\014b)s(er)f(of)h(the)g(semi-simpli\014cation)d(is)i (giv)m(en)h(b)m(y)g(the)h(w)m(a)m(ys)g(in)e(whic)m(h)g(this)g (irreducible)e(part)125 374 y(can)j(b)s(e)g(completed)h(with)e (o\013-diagonal)h(maps.)40 b(Decomp)s(ose)29 b(the)g(graded)f(pieces)h (of)f(the)h(t)m(yp)s(e)f(in)m(to)125 482 y(its)h(irreducible)e(comp)s (onen)m(ts)k(as)1262 651 y(Gr)1369 665 y Fx(i)1397 651 y FK(\()p FD(V)1485 665 y Fs(\017)1525 651 y FK(\))26 b(=)f FD(U)1744 665 y Fx(\027)1779 675 y Fq(i)p Fr(\000)p FG(1)1883 665 y FJ(+1)1998 651 y FC(\010)20 b FD(:)15 b(:)g(:)21 b FC(\010)f FD(U)2368 665 y Fx(\027)2403 675 y Fq(i)2448 651 y FD(;)125 821 y FK(for)32 b(an)g(increasing)e (sequence)j(0)c(=)f FD(\027)1417 835 y FJ(0)1484 821 y FD(<)g FC(\001)15 b(\001)g(\001)30 b FD(<)e(\027)1862 835 y Fx(i)p Fs(\000)p FJ(1)2008 821 y FD(<)g(\027)2152 835 y Fx(i)2209 821 y FD(<)g FC(\001)15 b(\001)g(\001)48 b FK(This)31 b(means)h(that)h(the)f(isot)m(ypic)125 928 y(comp)s(onen)m(ts)26 b FD(W)707 942 y Fx(i;j)813 928 y FK(will)e(b)s(e)i(isomorphic)e(to)j FD(m)1744 942 y Fx(i;j)1850 928 y FK(of)g(the)g(pieces)f FD(U)2424 942 y Fx(\013)2473 928 y FK(,)i(for)e FD(\027)2706 942 y Fx(i)p Fs(\000)p FJ(1)2849 928 y FD(<)f(\013)h FC(\024)f FD(\027)3170 942 y Fx(i)3198 928 y FK(.)39 b(Consider)125 1036 y FD(\045)k FK(=)330 968 y Ft(L)446 1036 y FD(\045)493 1050 y Fx(\013)587 1036 y FC(2)g(I)7 b FK(\()p FD(\034)j FK(\),)45 b(with)40 b FD(\045)1203 1050 y Fx(\013)1294 1036 y FK(the)i(irreducible)c(action)k(on)f FD(U)2406 1050 y Fx(\013)2456 1036 y FK(.)74 b(When)41 b(restricted)g(to)i Fz(Z)3435 1050 y Fx(m)3541 1036 y FC(\032)125 1144 y Fz(Z)190 1158 y Fx(n)257 1144 y FD(?)25 b Fz(Z)392 1158 y Fx(m)454 1144 y FK(,)38 b(the)f Fz(Z)745 1158 y Fx(m)808 1144 y FK(-mo)s(dule)e FD(U)1227 1158 y Fx(\013)1313 1144 y FK(is)h(semi-simple)e(with)i(1-dimensional)f(irreducible)e (pieces.)60 b(Let)125 1252 y FD(u)177 1266 y Fx(\013;)p FJ(1)281 1252 y FD(;)15 b(:)g(:)g(:)i(;)e(u)535 1267 y Fx(\013;d)636 1275 y Fq(\013)712 1252 y FK(b)s(e)27 b(a)g(basis)f(of)i FD(U)1288 1266 y Fx(\013)1337 1252 y FK(,)h(suc)m(h)e(that)g FC(h)p FD(u)1873 1267 y Fx(\013;l)1965 1252 y FC(i)g FK(is)g(an)g(irreducible)d Fz(Z)2751 1266 y Fx(m)2813 1252 y FK(-represen)m(tation)k(with)125 1360 y(c)m(haracter)k FD(")564 1375 y Fx(\013;l)655 1360 y FK(.)224 1504 y(No)m(w,)f(consider)f(the)g(v)m(ector)i(space)1359 1687 y FC(M)1468 1701 y FJ(0)1533 1687 y FK(=)1632 1600 y Ft(M)1629 1798 y Fx(\013<\014)1787 1687 y FK(Hom)1977 1702 y Fx(k)2020 1687 y FK(\()p FD(U)2117 1702 y Fx(\014)2164 1687 y FD(;)15 b(U)2266 1701 y Fx(\013)2316 1687 y FK(\))p FD(:)125 1953 y FK(Giv)m(en)28 b FD(M)35 b FC(2)25 b(M)704 1967 y FJ(0)744 1953 y FK(,)j(in)f(the)i(previous)e(basis)g(it)h(can)g (b)s(e)g(seen)g(as)h(a)f(collection)g(of)h(maps)f FD(M)3218 1968 y Fx(\013\014)3336 1953 y FK(:)d FD(U)3448 1968 y Fx(\014)3521 1953 y FC(!)125 2061 y FD(U)187 2075 y Fx(\013)236 2061 y FK(.)41 b(Let)31 b(us)e(consider)h(the)g(induced)f (represen)m(tation)h FD(\045)2072 2075 y Fx(M)2176 2061 y FK(:)c Fz(Z)16 b FD(?)k Fz(Z)h FC(!)k FK(GL)2703 2075 y Fx(r)2772 2061 y FK(b)m(y)528 2244 y FD(\045)575 2258 y Fx(M)654 2244 y FK(\()p FD(x)p FK(\)\()p FD(u)863 2259 y Fx(\014)s(;l)953 2244 y FK(\))h(=)e FD(\045)1156 2259 y Fx(\014)1204 2244 y FK(\()p FD(x)p FK(\)\()p FD(u)1413 2259 y Fx(\014)s(;l)1502 2244 y FK(\))p FD(;)107 b(\045)1716 2258 y Fx(M)1795 2244 y FK(\()p FD(y)s FK(\)\()p FD(u)2000 2259 y Fx(\014)s(;l)2090 2244 y FK(\))25 b(=)g FD(")2288 2259 y Fx(\014)s(;l)2392 2244 y FD(u)2444 2259 y Fx(\014)s(;l)2553 2244 y FK(+)2644 2157 y Ft(X)2687 2348 y Fx(\013)2790 2244 y FD(M)2878 2259 y Fx(\013\014)2971 2244 y FK(\()p FD(u)3058 2259 y Fx(\014)s(;l)3147 2244 y FK(\))p FD(:)125 2495 y FK(Here)f FD(x)f FK(and)g FD(y)j FK(denote)e(the)g(generators)g (of)g(eac)m(h)g(of)g(the)g(factors)g(of)g Fz(Z)6 b FD(?)g Fz(Z)n FK(.)32 b(Observ)m(e)24 b(that)g FD(\045)3267 2509 y Fx(M)3346 2495 y FK(\()p FD(x)p FK(\))3468 2462 y Fx(n)3541 2495 y FK(=)125 2535 y Ft(L)226 2630 y Fx(\014)288 2603 y FD(\045)335 2618 y Fx(\014)382 2603 y FK(\()p FD(x)p FK(\))504 2570 y Fx(n)577 2603 y FK(=)h(Id)o(,)30 b(but)e FD(\045)1023 2617 y Fx(M)1102 2603 y FK(\()p FD(y)s FK(\))1220 2570 y Fx(m)1316 2603 y FK(ma)m(y)i(not)f(b)s(e)g (the)g(iden)m(tit)m(y)g(map.)40 b(Hence,)30 b FD(\045)2863 2617 y Fx(M)2971 2603 y FK(can)g(b)s(e)e(seen)i(as)f(a)125 2714 y(represen)m(tation)h FD(\045)766 2728 y Fx(M)870 2714 y FK(:)c Fz(Z)986 2728 y Fx(n)1049 2714 y FD(?)20 b Fz(Z)h FC(!)k FK(GL)1444 2728 y Fx(r)1482 2714 y FK(.)224 2857 y(In)i(other)i(w)m(ords,)f(in)e(a)j(matrix)e(notation)h(in)f(the)h (basis)f(ab)s(o)m(v)m(e,)i(the)g(represen)m(tation)f(is)f(giv)m(en)g(b) m(y)125 2965 y(a)36 b(matrix)g FD(A)g FK(with)f(diagonal)h(blo)s(c)m (ks)g FD(\045)1524 2980 y Fx(\014)1571 2965 y FK(\()p FD(x)p FK(\),)i(and)e(b)m(y)g(a)h(matrix)f FD(B)41 b FK(whose)36 b(diagonal)f(blo)s(c)m(ks)h(are)125 3073 y(diagonal)e(matrices)h FD(B)931 3088 y Fx(\014)1011 3073 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FK(of)h(admissible)d(o\013-diagonal)i(maps,)i (i.e.)f FD(M)35 b FC(2)25 b(M)3572 3729 y FJ(0)125 3823 y FK(suc)m(h)30 b(that)h FD(\045)574 3837 y Fx(M)683 3823 y FK(is)e(a)i Fz(Z)915 3837 y Fx(n)978 3823 y FD(?)20 b Fz(Z)1108 3837 y Fx(m)1171 3823 y FK(-represen)m(tation.)224 3966 y(This)31 b(space)i(is)f(actually)g(a)h(linear)e(subspace)h(that)h (can)g(b)s(e)f(easily)f(c)m(haracterized.)49 b(F)-8 b(or)33 b FD(\013)c(<)g(\014)5 b FK(,)125 4074 y(write)29 b(in)m(to)h(its)g (comp)s(onen)m(ts)1509 4228 y FD(M)1597 4243 y Fx(\013\014)1715 4228 y FK(=)1811 4127 y Ft(\020)1865 4228 y FD(m)1945 4184 y Fx(ij)1945 4258 y(\013\014)2037 4127 y Ft(\021)2092 4292 y Fx(i;j)2202 4228 y FD(:)125 4419 y FK(W)-8 b(e)31 b(ha)m(v)m(e)h(the)e(follo)m(wing)f(c)m(haracterization.)125 4594 y SDict begin H.S end 125 4594 a 125 4594 a SDict begin 13 H.A end 125 4594 a 125 4594 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.1) cvn /DEST pdfmark end 125 4594 a 0 TeXcolorgray FL(Prop)s(osition)39 b(4.1.)p 0 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Fx(\015)1151 5430 y FG(1)1186 5421 y Fs(\024)p Fx(:::)o Fs(\024)p Fx(\015)1391 5430 y Fq(m)p Fr(\000)p FG(1)1528 5421 y Fs(\024)p Fx(\014)1641 5310 y FD(M)1729 5324 y Fx(\013\015)1810 5333 y FG(1)1850 5310 y FD(M)1938 5324 y Fx(\015)1974 5333 y FG(1)2009 5324 y Fx(\015)2045 5333 y FG(2)2099 5310 y FC(\001)15 b(\001)g(\001)h FD(M)2308 5325 y Fx(\015)2344 5334 y Fq(m)p Fr(\000)p FG(1)2482 5325 y Fx(\014)2554 5310 y FK(=)25 b(0)p FD(;)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.12) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(12)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 125 266 a FK(where)29 b(w)m(e)i(write)f FD(M)842 281 y Fx(\014)s(\014)957 266 y FK(=)25 b FD(B)1122 281 y Fx(\014)1170 266 y FK(.)40 b(Lo)s(oking)30 b(at)h(the)g(terms)f(con)m(taining)g FD(M)2636 281 y Fx(\013\014)2728 266 y FK(,)h(w)m(e)g(ha)m(v)m(e)125 496 y(\(14\))1356 496 y SDict begin H.S end 1356 496 a 1356 496 a SDict begin 13 H.A end 1356 496 a 1356 496 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.14) cvn /DEST pdfmark end 1356 496 a 1406 382 a Fx(m)1371 410 y Ft(X)1379 604 y Fx(t)p FJ(=1)1518 496 y FD(B)1592 459 y Fx(t)p Fs(\000)p FJ(1)1587 519 y Fx(\013)1711 496 y FD(M)1799 511 y Fx(\013\014)1892 496 y FD(B)1966 458 y Fx(m)p Fs(\000)p Fx(t)1961 525 y(\014)2137 496 y FK(=)25 b FD(M)2331 459 y Fs(0)2355 496 y FD(;)125 734 y FK(where)41 b FD(M)497 701 y Fs(0)561 734 y FK(is)g(a)h(\014xed)f(matrix)f(dep)s(ending)f(on)j (the)f(matrices)h 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FK(app)s(ear)24 b(in)g(consecutiv)m(e)h(steps)g (in)f(the)h(semi-simple)d(\014ltration\),)k(w)m(e)f(ha)m(v)m(e)h(that)g (the)f(righ)m(t)f(hand)125 1758 y(side)29 b(of)h(\()445 1758 y SDict begin H.S end 445 1758 a 0 TeXcolorgray FK(14)p 0 TeXcolorgray 537 1699 a SDict begin H.R end 537 1699 a 537 1758 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.14) cvn H.B /ANN pdfmark end 537 1758 a FK(\))g(v)-5 b(anishes.)40 b(Then)29 b FD(m)1312 1714 y Fx(ij)1312 1787 y(\013\014)1430 1758 y FK(=)c(0)30 b(when)g FD(")1881 1772 y Fx(\013;i)2000 1758 y FK(=)24 b FD(")2137 1773 y Fx(\014)s(;j)2237 1758 y FK(.)125 1934 y SDict begin H.S end 125 1934 a 125 1934 a SDict begin 13 H.A end 125 1934 a 125 1934 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.3) cvn /DEST pdfmark end 125 1934 a 0 TeXcolorgray FL(Lemma)f(4.3.)p 0 TeXcolorgray 36 w Fp(A)n(ny)i(r)-5 b(epr)g(esentation)28 b FD(\032)d FC(2)g FD(R)q FK(\()p FD(\034)10 b FK(\))26 b 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FD(R)q FK(\()p FD(\034)10 b FK(\))38 b FC(!)f(I)7 b FK(\()p FD(\034)j FK(\))38 b(is)f(the)h (semi-simpli\014cation)c(map)j(and)g FD(\045)g FC(2)h(I)7 b FK(\()p FD(\034)j FK(\))125 3091 y(then)29 b(the)h(\014b)s(er)f(Gr) 804 3055 y Fs(\000)p FJ(1)804 3114 y Fs(\017)898 3091 y FK(\()p FD(\045)p FK(\))i(is)e(con)m(tained)h(in)f FC(M)1763 3105 y FJ(1)1803 3091 y FK(.)40 b(Ho)m(w)m(ev)m(er,)33 b(this)28 b(\014bre)h(ma)m(y)i(not)f(b)s(e)g(the)g(whole)f(of)125 3199 y FC(M)234 3213 y FJ(1)273 3199 y FK(,)34 b(since)f(an)g(elemen)m (t)h FD(M)40 b FC(2)30 b(M)1354 3213 y FJ(1)1427 3199 y FK(ma)m(y)k(not)g(preserv)m(e)f(the)h(non-splitting)c(conditions)i (imp)s(osed)125 3307 y(b)m(y)k(the)h(semi-simple)d(\014ltration.)58 b(F)-8 b(or)37 b(instance,)h(if)d FD(M)2083 3322 y Fx(\013\014)2211 3307 y FK(=)h(0)h(for)f(a)h(\014xed)f FD(\027)2898 3321 y Fx(i)2961 3307 y FD(<)f(\014)41 b FC(\024)35 b FD(\027)3310 3321 y Fx(i)p FJ(+1)3465 3307 y FK(and)125 3415 y(for)30 b(all)g FD(\027)436 3429 y Fx(i)p Fs(\000)p FJ(1)581 3415 y FD(<)c(\013)h FC(\024)f FD(\027)905 3429 y Fx(i)964 3415 y FK(then)31 b FD(i)p FK(-th)g(step)g(of)g(the)h(semi-simple)c (\014ltration)i(is)g(not)h(maximal,)f(since)h(it)125 3523 y(can)k(b)s(e)f(extended)h(with)e FD(U)1088 3538 y Fx(\014)1135 3523 y FK(.)54 b(Hence,)37 b(the)e(c)m(haracterization)h (of)f(this)f(\014b)s(er)f(will)f(require)i(further)125 3631 y(analysis.)224 3774 y(Fix)c FD(\034)35 b FC(2)25 b(T)598 3788 y Fx(\024)673 3774 y FK(with)k FD(\024)d FK(=)e(\()p Fn(\017)p FD(;)15 b Fn(")p FK(\).)42 b(Consider)29 b(the)h(`unsymmetrized')f(v)m(ersions)h(of)h(\()3027 3774 y SDict begin H.S end 3027 3774 a 0 TeXcolorgray FK(13)p 0 TeXcolorgray 3118 3716 a SDict begin H.R end 3118 3716 a 3118 3774 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.13) cvn H.B /ANN pdfmark end 3118 3774 a FK(\))g(giv)m(en)f(b)m (y)1354 3979 y(^)1346 4002 y FC(I)6 b FK(\()p FD(\034)k FK(\))26 b(=)1686 3888 y Fx(s)1644 3915 y Ft(Y)1645 4111 y Fx(i)p FJ(=1)1807 3886 y Fx(s)1840 3896 y Fq(i)1779 3915 y Ft(Y)1776 4111 y Fx(j)t FJ(=1)1913 3901 y Ft(\020)1968 4002 y FD(R)2038 3964 y FJ(irr)2037 4024 y Fx(\024)2078 4034 y Fq(i;j)2155 3901 y Ft(\021)2209 3923 y Fx(m)2271 3933 y Fq(i;j)2365 4002 y FD(:)125 4270 y FK(There)j(is)h(a)h(quotien)m (t)f(map)1123 4247 y(^)1115 4270 y FC(I)6 b FK(\()p FD(\034)k FK(\))26 b FC(!)g(I)7 b FK(\()p FD(\034)j FK(\).)40 b(Consider)29 b(the)h(pullbac)m(k)f(diagram)125 4457 y(\(15\))1470 4457 y SDict begin H.S end 1470 4457 a 1470 4457 a SDict begin 13 H.A end 1470 4457 a 1470 4457 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.15) cvn /DEST pdfmark end 1470 4457 a 1568 4434 a FK(^)1548 4457 y FD(R)q FK(\()p FD(\034)10 b FK(\))1971 4435 y Fw(/)p Fv(/)p 1765 4436 206 4 v 1521 4626 a FJ(^)1497 4643 y(Gr)1580 4651 y Fr(\017)1643 4706 y Fw(\017)p Fv(\017)p 1642 4706 4 200 v 1998 4457 a FD(R)q FK(\()p FD(\034)g FK(\))2120 4634 y FJ(Gr)2203 4643 y Fr(\017)2093 4724 y Fw(\017)p Fv(\017)p 2091 4724 4 218 v 1563 4796 a FK(^)1555 4819 y FC(I)c FK(\()p FD(\034)k FK(\))1965 4797 y Fw(/)p Fv(/)p 1759 4798 207 4 v 1992 4819 a FC(I)d FK(\()p FD(\034)j FK(\))p FD(:)224 5042 y FK(No)m(w)30 b(w)m(e)g(shall)e(study)1044 5019 y(^)1025 5042 y FD(R)p FK(\()p FD(\034)10 b FK(\))31 b(in)d(detail.)39 b(F)-8 b(or)31 b(that)f(purp)s(ose,)e(let)h(us)g(consider)g(the)g(map) 3326 5019 y(^)3318 5042 y FC(I)7 b FK(\()p FD(\034)j FK(\))26 b FC(!)125 5150 y FK(GL)253 5164 y Fx(r)330 5150 y FK(giv)m(en)39 b(b)m(y)g FD(\045)g FK(=)h(\()p FD(A;)15 b(B)5 b FK(\))40 b FC(7!)g FD(B)5 b FK(.)66 b(The)39 b(\014b)s(er)e(o)m(v)m(er)j(\006)2179 5164 y Fm(")2260 5150 y FC(2)f FK(GL)2488 5164 y Fx(r)2526 5150 y FK(,)j(the)d(diagonal)f(matrix)h(with)125 5258 y(eigen)m(v)-5 b(alues)30 b Fn(")p FK(,)g(will)e(b)s(e)i(denoted)g(b)m(y)125 5413 y(\(16\))1266 5413 y SDict begin H.S end 1266 5413 a 1266 5413 a SDict begin 13 H.A end 1266 5413 a 1266 5413 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.16) cvn /DEST pdfmark end 1266 5413 a 1274 5390 a FK(^)1266 5413 y FC(I)1316 5427 y FJ(0)1355 5413 y FK(\()p FD(\034)10 b FK(\))26 b(=)1597 5339 y Ft(\010)1650 5413 y FD(\045)f FK(=)g(\()p FD(A;)15 b FK(\006)2027 5427 y Fm(")2069 5413 y FK(\))25 b FC(2)2223 5390 y FK(^)2215 5413 y FC(I)6 b FK(\()p FD(\034)k FK(\))2391 5339 y Ft(\011)2445 5413 y FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.13) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(13)p 0 TeXcolorgray 125 266 a FK(These)32 b(are)g(semi-simple)e(represen)m(tations)j(whose)f (matrix)g FD(B)k FK(is)c(diagonal)f(\(not)i(just)f(diagonaliz-)125 374 y(able\).)224 526 y(Fix)e FD(\045)25 b FC(2)553 503 y FK(^)545 526 y FC(I)595 540 y FJ(0)634 526 y FK(\()p FD(\034)10 b FK(\).)41 b(Let)125 685 y(\(17\))1247 685 y SDict begin H.S end 1247 685 a 1247 685 a SDict begin 13 H.A end 1247 685 a 1247 685 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.17) cvn /DEST pdfmark end 1247 685 a FC(M)1356 699 y Fx(\045)1421 685 y FK(=)1517 612 y Ft(\010)1570 685 y FD(M)36 b FC(2)24 b(M)1888 699 y FJ(1)1943 685 y FC(j)15 b FD(\045)2030 699 y Fx(M)2135 685 y FC(2)2240 662 y FK(^)2221 685 y FD(R)p FK(\()p FD(\034)10 b FK(\))2410 612 y Ft(\011)2464 685 y FD(;)125 835 y FK(that)38 b(is,)g(the)g(set)g (of)g(admissible)d(upp)s(er-triangular)f(completions)j(of)g FD(\045)p FK(.)62 b(W)-8 b(e)39 b(also)f(consider)e(the)125 943 y(gauge)31 b(group)f(of)g FC(M)852 957 y Fx(\045)923 943 y FK(de\014ned)f(as)125 1101 y(\(18\))515 1101 y SDict begin H.S end 515 1101 a 515 1101 a SDict begin 13 H.A end 515 1101 a 515 1101 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.18) cvn /DEST pdfmark end 515 1101 a FC(G)569 1115 y Fx(\045)635 1101 y FK(=)c FC(f)p FD(P)39 b FC(2)25 b FK(GL)1087 1115 y Fx(r)1155 1101 y FC(j)15 b FK(for)31 b(all)e FD(M)35 b FC(2)25 b(M)1779 1115 y Fx(\045)1820 1101 y FD(;)15 b(P)e(\045)1978 1115 y Fx(M)2057 1101 y FD(P)2128 1063 y Fs(\000)p FJ(1)2247 1101 y FK(=)25 b FD(\045)2390 1116 y Fx(M)2465 1097 y Fr(0)2492 1101 y FD(;)46 b FK(for)30 b(some)g FD(M)3027 1063 y Fs(0)3076 1101 y FC(2)25 b(M)3271 1115 y Fx(\045)3311 1101 y FC(g)p FD(:)125 1265 y SDict begin H.S end 125 1265 a 125 1265 a SDict begin 13 H.A end 125 1265 a 125 1265 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.4) cvn /DEST pdfmark end 125 1265 a 0 TeXcolorgray FL(Theorem)48 b(4.4.)p 0 TeXcolorgray 50 w FC(M)904 1279 y Fx(\045)989 1265 y Fp(is)c(an)h(algebr)-5 b(aic)46 b(variety)f(and)h FC(G)2188 1279 y Fx(\045)2273 1265 y Fp(is)e(an)h(algebr)-5 b(aic)46 b(sub)-5 b(gr)g(oup)46 b(of)e FK(GL)3546 1279 y Fx(r)3584 1265 y Fp(.)125 1373 y(Mor)-5 b(e)g(over,)35 b(they)f(ar)-5 b(e)35 b(indep)-5 b(endent)35 b(of)f(the)g(r)-5 b(epr)g(esentation,)37 b(in)c(the)h(sense)g(that)h FC(M)3096 1387 y Fx(\045)3164 1348 y FC(\030)3164 1377 y FK(=)3262 1373 y FC(M)3371 1388 y Fx(\045)3407 1369 y Fr(0)3467 1373 y Fp(and)125 1495 y FC(G)179 1509 y Fx(\045)244 1470 y FC(\030)244 1500 y FK(=)340 1495 y FC(G)394 1511 y Fx(\045)430 1492 y Fr(0)489 1495 y Fp(for)f(any)f FD(\045;)15 b(\045)943 1462 y Fs(0)992 1495 y FC(2)1086 1472 y FK(^)1078 1495 y FC(I)1128 1509 y FJ(0)1167 1495 y FK(\()p FD(\034)10 b FK(\))p Fp(.)p 0 TeXcolorgray 125 1694 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(Once)23 b(w)m(e)f(kno)m(w)g(that)h FC(M)1282 1708 y Fx(\045)1344 1694 y FK(is)f(an)g(algebraic)f(v)-5 b(ariet)m(y)d(,)25 b(then)d(the)g(fact)h(that)g FC(G)2994 1708 y Fx(\045)3057 1694 y FK(is)e(an)h(algebraic)125 1802 y(subgroup)28 b(follo)m(ws)i(easily)-8 b(.)40 b(Indeed,)30 b(consider)f(the)i(cartesian)f(square)1494 1959 y FD(Z)1829 1937 y Fw(/)p Fv(/)p 1589 1938 240 4 v 1528 2212 a Fw(\017)p Fv(\017)p 1526 2212 4 226 v 1856 1959 a FC(M)1965 1973 y Fx(\045)2026 1959 y FC(\002)20 b FK(GL)2245 1973 y Fx(r)2069 2212 y Fw(\017)p Fv(\017)p 2068 2212 4 200 v 1453 2301 a FC(M)1562 2315 y Fx(\045)1948 2279 y Fw(/)p Fv(/)p 1630 2280 319 4 v 1975 2301 a FC(M)2084 2315 y FJ(1)2139 2301 y FD(;)125 2473 y FK(where)32 b(the)h(b)s(ottom)h (morphism)d(is)h(the)h(inclusion)d(and)j(the)g(righ)m(tmost)g(map)g(is) f(\()p FD(P)s(;)15 b(\045)3172 2487 y Fx(M)3252 2473 y FK(\))30 b FC(7!)g(M)3547 2487 y FJ(1)3586 2473 y FK(.)125 2581 y(The)f(pullbac)m(k)g(v)-5 b(ariet)m(y)31 b(is)e(th)m(us)969 2731 y FD(Z)j FK(=)1159 2658 y Ft(\010)1212 2731 y FK(\()p FD(P)s(;)15 b(M)10 b FK(\))26 b FC(2)f FK(GL)1722 2745 y Fx(r)1775 2731 y FC(\002M)1955 2745 y Fx(\045)2010 2731 y FC(j)15 b FD(P)e(\045)2168 2745 y Fx(M)2247 2731 y FD(P)2318 2694 y Fs(\000)p FJ(1)2438 2731 y FC(2)25 b(M)2633 2745 y Fx(\045)2673 2658 y Ft(\011)2741 2731 y FD(:)125 2882 y FK(By)37 b(de\014nition,)g FC(G)766 2896 y Fx(\045)844 2882 y FK(=)g FD(\031)s FK(\()p FD(Z)7 b FK(\))37 b(where)g FD(\031)j FK(:)e(GL)1736 2896 y Fx(r)1789 2882 y FC(\002M)1969 2896 y Fx(\045)2046 2882 y FC(!)f FK(GL)2302 2896 y Fx(r)2378 2882 y FK(is)f(the)i(pro)5 b(jection)37 b(on)m(to)h(the)g(\014rst)125 2990 y(comp)s(onen)m(t.)h (The)26 b(pro)5 b(jection)25 b(of)h(an)g(algebraic)g(v)-5 b(ariet)m(y)26 b(is)f(a)i(constructible)e(set)h(\(a)h(disjoin)m(t)d (union)125 3098 y(of)j(lo)s(cally)g(closed)g(subsets)g(in)g(the)h (Zariski)e(top)s(ology\).)40 b(As)28 b FC(G)2274 3112 y Fx(\045)2342 3098 y FK(is)f(a)h(group,)g(it)f(is)g(an)h(homogeneous) 125 3205 y(space.)38 b(Hence)23 b(taking)f(a)h(smo)s(oth)f(p)s(oin)m(t) f(whic)m(h)g(has)i(an)f(op)s(en)f(neigh)m(b)s(ourho)s(o)s(d)f(where)h FC(G)3202 3219 y Fx(\045)3265 3205 y FK(is)g(closed,)125 3313 y(w)m(e)31 b(see)h(that)g(this)e(happ)s(ens)g(at)i(ev)m(ery)g(p)s (oin)m(t.)43 b(Therefore)31 b FC(G)2237 3327 y Fx(\045)2308 3313 y FK(is)g(quasi-pro)5 b(jectiv)m(e,)31 b(and)g(hence)g(an)125 3421 y(algebraic)f(v)-5 b(ariet)m(y)d(.)224 3565 y(With)36 b(resp)s(ect)g(to)h FC(M)1008 3579 y Fx(\045)1049 3565 y FK(,)h(tak)m(e)g FD(M)45 b FK(=)35 b FC(\010)15 b FD(M)1728 3580 y Fx(\013\014)1855 3565 y FC(2)35 b(M)2060 3579 y FJ(1)2100 3565 y FK(.)58 b(In)36 b(order)f(to)i(lie)e(in)g FC(M)3011 3579 y Fx(\045)3052 3565 y FK(,)j(the)e(matrices)125 3673 y FD(M)213 3688 y Fx(\013\014)337 3673 y FK(m)m(ust)c(satisfy)g(t) m(w)m(o)i(conditions.)44 b(In)32 b(the)g(\014rst)f(place,)i(if)e FD(\027)2352 3687 y Fx(i)p Fs(\000)p FJ(1)2499 3673 y FD(<)d(\013)g(<)g(\014)34 b FC(\024)28 b FD(\027)3012 3687 y Fx(i)3072 3673 y FK(then)k(w)m(e)h(m)m(ust)125 3781 y(ha)m(v)m(e)e FD(M)422 3796 y Fx(\013\014)540 3781 y FK(=)25 b(0.)41 b(Th)m(us)29 b(w)m(e)i(set)g(the)g(linear)e(subspace) 448 3936 y FC(H)d FK(=)647 3862 y Ft(\010)700 3936 y FD(M)35 b FK(=)25 b FC(\010)15 b FD(M)1093 3951 y Fx(\013\014)1201 3936 y FC(j)g FD(M)1329 3951 y Fx(\013\014)1447 3936 y FK(=)25 b(0)p FD(;)46 b FK(if)29 b FD(\027)1787 3950 y Fx(i)p Fs(\000)p FJ(1)1931 3936 y FD(<)c(\013)h(<)f(\014)30 b FC(\024)25 b FD(\027)2429 3950 y Fx(i)2457 3936 y FD(;)15 b(i)26 b FK(=)f(2)p FD(;)15 b(:)g(:)g(:)i(;)e(s)2940 3862 y Ft(\011)3018 3936 y FC(\032)25 b(M)3223 3950 y FJ(1)3263 3936 y FD(:)224 4121 y FK(Second,)34 b(if)f FD(\027)693 4135 y Fx(i)p Fs(\000)p FJ(1)842 4121 y FD(<)d(\013)h FC(\024)f FD(\027)1178 4135 y Fx(i)1239 4121 y FK(and)j FD(\027)1464 4135 y Fx(i)1523 4121 y FD(<)d(\014)35 b FC(\024)c FD(\027)1857 4135 y Fx(i)p FJ(+1)1975 4121 y FK(,)j(then)f(the)h(maximalit)m(y)f(of)g(the)h(semi-simple)125 4229 y(sub-represen)m(tations)29 b(implies)f(that)j(w)m(e)f(cannot)h (ha)m(v)m(e)h(a)f(larger)f(decomp)s(osition)125 4379 y(\(19\))1326 4379 y SDict begin H.S end 1326 4379 a 1326 4379 a SDict begin 13 H.A end 1326 4379 a 1326 4379 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.19) cvn /DEST pdfmark end 1326 4379 a 15 w FK(Gr)1448 4393 y Fx(i)p Fs(\000)p FJ(1)1566 4379 y FK(\()p FD(\045)1648 4393 y Fx(M)1728 4379 y FK(\))20 b FC(\010)g FD(U)1936 4394 y Fx(\014)2009 4379 y FC(\032)25 b FD(V)10 b(=V)2266 4393 y Fx(i)p Fs(\000)p FJ(1)2385 4379 y FD(:)125 4529 y FK(W)-8 b(e)37 b(ha)m(v)m(e)g(a)f(decomp)s(osition)f(\()1223 4529 y SDict begin H.S end 1223 4529 a 0 TeXcolorgray FK(19)p 0 TeXcolorgray 1314 4471 a SDict begin H.R end 1314 4471 a 1314 4529 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.19) cvn H.B /ANN pdfmark end 1314 4529 a FK(\))h(when)f(there)h (is)f(a)h(c)m(hange)h(of)f(basis)f(suc)m(h)h(that)g FD(M)3189 4544 y Fx(\013\014)3316 4529 y FK(=)e(0)j(for)125 4637 y(all)29 b FD(\013)d FK(=)g FD(\027)477 4651 y Fx(i)p Fs(\000)p FJ(1)616 4637 y FK(+)20 b(1)p FD(;)15 b(:)g(:)g(:)i(;)e(\027) 999 4651 y Fx(i)1027 4637 y FK(.)42 b(Let)31 b(us)f(\014x)g FD(\013;)15 b(\014)37 b FK(and)30 b(let)h(us)f(only)f(fo)s(cus)i(on)f (these)h(blo)s(c)m(ks.)41 b(There,)31 b(w)m(e)125 4745 y(ha)m(v)m(e)g(that)g FD(\045)25 b FK(=)g(\()p FD(A;)15 b(B)5 b FK(\))31 b(and)f FD(\045)1206 4759 y Fx(M)1310 4745 y FK(=)25 b(\()p FD(A;)15 b(B)1618 4759 y Fx(M)1698 4745 y FK(\),)31 b(where)666 4950 y FD(A)25 b FK(=)855 4822 y Ft(\022)922 4895 y FD(A)990 4909 y Fx(\013)1158 4895 y FK(0)958 5003 y(0)120 b FD(A)1191 5018 y Fx(\014)1238 4822 y Ft(\023)1320 4950 y FD(;)107 b(B)30 b FK(=)1647 4822 y Ft(\022)1713 4895 y FD(B)1782 4909 y Fx(\013)1950 4895 y FK(0)1750 5003 y(0)120 b FD(B)1984 5018 y Fx(\014)2031 4822 y Ft(\023)2113 4950 y FD(;)106 b(B)2313 4964 y Fx(M)2418 4950 y FK(=)2514 4822 y Ft(\022)2580 4895 y FD(B)2649 4909 y Fx(\013)2782 4895 y FD(M)2870 4910 y Fx(\013\014)2617 5003 y FK(0)152 b FD(B)2883 5018 y Fx(\014)2963 4822 y Ft(\023)3045 4950 y FD(;)125 5150 y FK(where)28 b(the)g(matrices)h FD(B)972 5164 y Fx(\013)1021 5150 y FD(;)15 b(B)1130 5165 y Fx(\014)1206 5150 y FK(are)29 b(diagonal.)40 b(Th)m(us)27 b(a)i(c)m(hange)h(of)f(basis)e FD(P)42 b FK(has)28 b(to)h(b)s(e)f(of)h (the)g(form)1537 5360 y FD(P)38 b FK(=)1729 5232 y Ft(\022)1796 5305 y FD(P)1854 5319 y Fx(\013)2009 5305 y FD(S)1827 5413 y FK(0)115 b FD(P)2045 5428 y Fx(\014)2092 5232 y Ft(\023)2174 5360 y FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.14) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(14)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 125 266 a FK(Since)43 b FD(P)13 b(\045)494 280 y Fx(M)573 266 y FD(P)644 233 y Fs(\000)p FJ(1)783 266 y FK(has)45 b(to)g(lie)f(in)f FC(M)1450 280 y Fx(\045)1491 266 y FK(,)48 b(w)m(e)e(m)m(ust)e(ha)m(v)m (e)i FD(P)13 b(AP)2386 233 y Fs(\000)p FJ(1)2530 266 y FK(=)49 b FD(A)p FK(,)g(and)44 b(this)g(forces)h(that)125 374 y FD(P)183 388 y Fx(\013)261 374 y FK(=)28 b FD(\025)413 388 y Fx(\013)463 374 y FK(Id)o(,)33 b FD(P)662 389 y Fx(\014)738 374 y FK(=)28 b FD(\025)890 389 y Fx(\014)938 374 y FK(Id)j(for)h(some)h FD(\025)1477 388 y Fx(\013)1527 374 y FD(;)15 b(\025)1620 389 y Fx(\014)1696 374 y FC(2)28 b FD(k)1835 341 y Fs(\003)1874 374 y FK(.)47 b(No)m(w,)34 b(w)m(e)f(consider)e(a)i(c)m(hange)g(of)g(co)s(ordinates)125 482 y(of)d(the)h(form)125 663 y(\(20\))1523 663 y SDict begin H.S end 1523 663 a 1523 663 a SDict begin 13 H.A end 1523 663 a 1523 663 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.20) cvn /DEST pdfmark end 1523 663 a FD(Q)25 b FK(=)1716 535 y Ft(\022)1783 608 y FD(Q)1855 622 y Fx(\013)2024 608 y FK(0)1821 716 y(0)121 b FD(Q)2059 731 y Fx(\014)2106 535 y Ft(\023)2188 663 y FD(;)125 863 y FK(suc)m(h)30 b(that)761 1008 y FD(QAQ)973 970 y Fs(\000)p FJ(1)1092 1008 y FK(=)1188 879 y Ft(\022)1255 951 y FK(diag)q(\()p FD(\017)1495 965 y Fx(\013;)p FJ(1)1600 951 y FD(;)15 b(:)g(:)g(:)h(;)f(\017)1838 966 y Fx(\013;d)1939 974 y Fq(\013)1988 951 y FK(\))442 b(0)1617 1058 y(0)j(diag)q(\()p FD(\017)2347 1073 y Fx(\014)s(;)p FJ(1)2449 1058 y FD(;)15 b(:)g(:)g(:)i(;)e(\017)2688 1073 y Fx(\014)s(;d)2787 1085 y Fq(\014)2832 1058 y FK(\))2868 879 y Ft(\023)2950 1008 y FD(:)125 1207 y FK(It)23 b(is)f(straigh)m(tforw)m(ard)h(to)h(c)m (hec)m(k)h(that)f FD(S)30 b FK(=)25 b FD(Q)1719 1221 y Fx(\013)1768 1207 y FK(\002)p FD(Q)1911 1169 y Fs(\000)p FJ(1)1911 1236 y Fx(\014)2005 1207 y FK(,)g(where)e(\002)i(=)g(\()p FD(\022)2581 1221 y Fx(ij)2642 1207 y FK(\))e(is)f(a)i(matrix)f(with)f FD(\022)3385 1221 y Fx(ij)3470 1207 y FK(=)j(0)125 1320 y(if)k FD(\017)245 1334 y Fx(\013;i)363 1320 y FC(6)p FK(=)c FD(\017)496 1335 y Fx(\014)s(;j)595 1320 y FK(.)41 b(In)29 b(this)h(fashion,)f(the)i(action)f(of)h FD(P)43 b FK(is)30 b(giv)m(en)g(b)m(y)701 1479 y FD(M)789 1494 y Fx(\013\014)907 1479 y FC(7!)15 b FD(\025)1066 1493 y Fx(\013)1116 1479 y FD(\025)1169 1441 y Fs(\000)p FJ(1)1169 1509 y Fx(\014)1263 1479 y FD(M)1351 1494 y Fx(\013\014)1464 1479 y FK(+)20 b FD(\025)1608 1441 y Fs(\000)p FJ(1)1608 1509 y Fx(\014)1717 1479 y FK(\()q FD(S)5 b(B)1883 1494 y Fx(\014)1950 1479 y FC(\000)20 b FD(B)2110 1493 y Fx(\013)2159 1479 y FD(S)5 b FK(\))1023 1661 y(=)25 b FD(\025)1172 1675 y Fx(\013)1222 1661 y FD(\025)1275 1623 y Fs(\000)p FJ(1)1275 1691 y Fx(\014)1369 1661 y FD(M)1457 1676 y Fx(\013\014)1570 1661 y FK(+)20 b FD(\025)1714 1623 y Fs(\000)p FJ(1)1714 1691 y Fx(\014)1824 1560 y Ft(\020)1878 1661 y FD(Q)1950 1675 y Fx(\013)1999 1661 y FK(\002)p FD(Q)2142 1623 y Fs(\000)p FJ(1)2142 1691 y Fx(\014)2236 1661 y FD(B)2305 1676 y Fx(\014)2372 1661 y FC(\000)g FD(B)2532 1675 y Fx(\013)2582 1661 y FD(Q)2654 1675 y Fx(\013)2703 1661 y FK(\002)p FD(Q)2846 1623 y Fs(\000)p FJ(1)2846 1691 y Fx(\014)2940 1560 y Ft(\021)3009 1661 y FD(:)125 1833 y FK(Observ)m(e)37 b(that)h(this)e(action)h(preserv)m (es)h(the)f(conditions)f(of)i(Prop)s(osition)2758 1833 y SDict begin H.S end 2758 1833 a 0 TeXcolorgray FK(4.1)p 0 TeXcolorgray 2874 1775 a SDict begin H.R end 2874 1775 a 2874 1833 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.1) cvn H.B /ANN pdfmark end 2874 1833 a FK(.)61 b(W)-8 b(e)39 b(consider)d(the)125 1941 y(linear)30 b(space)j FD(`)662 1956 y Fx(\013\014)783 1941 y FK(=)28 b FC(h)p FD(Q)989 1955 y Fx(\013)1039 1941 y FK(\002)p FD(Q)1182 1903 y Fs(\000)p FJ(1)1182 1971 y Fx(\014)1276 1941 y FD(B)1345 1956 y Fx(\014)1413 1941 y FC(\000)22 b FD(B)1575 1955 y Fx(\013)1624 1941 y FD(Q)1696 1955 y Fx(\013)1745 1941 y FK(\002)p FD(Q)1888 1903 y Fs(\000)p FJ(1)1888 1971 y Fx(\014)1982 1941 y FC(i)p FK(.)47 b(Observ)m(e)32 b(that)h FD(`)2677 1956 y Fx(\013\014)2802 1941 y FK(only)e(dep)s(ends) g(on)h(the)125 2055 y(base)e(represen)m(tation)h FD(\045)p FK(.)224 2198 y(In)37 b(this)g(w)m(a)m(y)-8 b(,)42 b(the)c(condition)e (that,)41 b(after)d(a)h(c)m(hange)g(of)f(v)-5 b(ariables,)38 b(the)h(blo)s(c)m(ks)e FD(M)3216 2213 y Fx(\013\014)3346 2198 y FK(do)h(not)125 2306 y(v)-5 b(anish)30 b(for)i(all)f FD(\027)721 2320 y Fx(i)p Fs(\000)p FJ(1)867 2306 y FD(<)d(\013)h FC(\024)f FD(\027)1197 2320 y Fx(i)1225 2306 y FK(,)33 b(is)e(that)h(it)g(cannot)h(happ)s(en)d(that)j FD(M)2576 2321 y Fx(\013\014)2697 2306 y FC(2)28 b FD(`)2824 2321 y Fx(\013\014)2948 2306 y FK(for)k(all)f FD(\013)p FK(.)46 b(Hence,)125 2414 y(w)m(e)30 b(tak)m(e)481 2564 y FD(L)543 2579 y Fx(\014)615 2564 y FK(=)820 2478 y Ft(M)711 2673 y Fx(\027)746 2683 y Fq(i)p Fr(\000)p FG(1)850 2673 y Fx(<\013)p Fs(\024)p Fx(\027)1040 2683 y Fq(i)1082 2564 y FD(`)1120 2579 y Fx(\013\014)1237 2564 y FC(\032)1442 2478 y Ft(M)1333 2673 y Fx(\027)1368 2683 y Fq(i)p Fr(\000)p FG(1)1473 2673 y Fx(<\013)p Fs(\024)p Fx(\027)1663 2683 y Fq(i)1704 2564 y FK(Hom)1893 2579 y Fx(k)1951 2564 y FK(\()q FD(U)2049 2579 y Fx(\014)2096 2564 y FD(;)15 b(U)2198 2578 y Fx(\013)2248 2564 y FK(\))25 b(=)g(Hom)2594 2579 y Fx(k)2652 2564 y FK(\()p FD(U)2749 2579 y Fx(\014)2796 2564 y FD(;)15 b FK(Gr)2943 2578 y Fx(i)p Fs(\000)p FJ(1)3062 2564 y FK(\()p FD(\045)p FK(\)\))h FD(:)224 2853 y FK(In)32 b(the)h(case)h(that)f(all)f(the)h(isot)m(ypic)f(comp)s(onen)m(ts)h(of)g (Gr)2233 2867 y Fx(i)2261 2853 y FK(\()p FD(\045)2343 2867 y Fx(M)2422 2853 y FK(\))h(are)f(of)g(m)m(ultiplicit)m(y)c(one)k (\(that)125 2961 y(is,)27 b FD(m)319 2975 y Fx(i;j)424 2961 y FK(=)e(1)j(for)f(all)f FD(j)5 b FK(\),)29 b(then)e(the)h(only)e (p)s(ossible)f(decomp)s(ositions)h(\()2543 2961 y SDict begin H.S end 2543 2961 a 0 TeXcolorgray FK(19)p 0 TeXcolorgray 2634 2903 a SDict begin H.R end 2634 2903 a 2634 2961 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.19) cvn H.B /ANN pdfmark end 2634 2961 a FK(\))h(m)m(ust)h(in)m(v)m(olv)m (e)f(one)h(of)f(the)125 3069 y FD(U)187 3084 y Fx(\014)234 3069 y FK(.)40 b(Therefore)31 b(the)f(condition)f(means)h(that)1246 3219 y(p)1297 3234 y Fx(\014)1344 3219 y FK(\()p FD(M)10 b FK(\))26 b(=)f(\()p FD(M)1757 3234 y Fx(\013\014)1850 3219 y FK(\))1885 3233 y Fx(\027)1920 3243 y Fq(i)p Fr(\000)p FG(1)2024 3233 y Fx(<\013)p Fs(\024)p Fx(\027)2214 3243 y Fq(i)2280 3219 y FD(=)-55 b FC(2)25 b FD(L)2418 3234 y Fx(\014)2465 3219 y FD(;)125 3368 y FK(for)30 b(all)f FD(\027)435 3382 y Fx(i)488 3368 y FD(<)c(\014)31 b FC(\024)25 b FD(\027)807 3382 y Fx(i)p FJ(+1)925 3368 y FK(.)41 b(So)30 b(setting)441 3518 y FC(L)25 b FK(=)625 3444 y Ft(\010)678 3518 y FD(M)35 b FK(=)25 b FC(\010)15 b FD(M)1071 3533 y Fx(\013\014)1179 3518 y FC(j)g FK(p)1270 3533 y Fx(\014)1317 3518 y FK(\()p FD(M)10 b FK(\))36 b FD(=)-55 b FC(2)25 b FD(L)1659 3533 y Fx(\014)1706 3518 y FD(;)46 b FK(for)30 b(all)f FD(\027)2087 3532 y Fx(i)2140 3518 y FD(<)c(\014)31 b FC(\024)25 b FD(\027)2459 3532 y Fx(i)p FJ(+1)2577 3518 y FD(;)15 b(i)26 b FK(=)f(1)p FD(;)15 b(:)g(:)g(:)i(;)e(s)20 b FC(\000)g FK(1)3216 3444 y Ft(\011)3270 3518 y FD(;)125 3686 y FK(w)m(e)25 b(ha)m(v)m(e)i(that)e FC(M)758 3700 y Fx(\045)824 3686 y FK(=)g FC(H)11 b(\\)f(L)p FK(.)37 b(Observ)m(e)25 b(that)h FC(H)g FK(is)e(indep)s(enden)m(t)f(of)i FD(\045)g FK(and,)h(for)f(an)m (y)h FD(\045;)15 b(\045)3242 3653 y Fs(0)3291 3686 y FC(2)3384 3663 y FK(^)3377 3686 y FC(I)3427 3700 y FJ(0)3465 3686 y FK(\()p FD(\034)10 b FK(\),)125 3794 y(w)m(e)30 b(ha)m(v)m(e)i(that)f FC(L)753 3769 y(\030)753 3798 y FK(=)849 3794 y FC(L)912 3761 y Fs(0)935 3794 y FK(.)41 b(Hence,)31 b(w)m(e)g(ha)m(v)m(e)g(that)g FC(M)1945 3808 y Fx(\045)2011 3769 y FC(\030)2011 3798 y FK(=)2107 3794 y FC(M)2216 3809 y Fx(\045)2252 3790 y Fr(0)2279 3794 y FK(.)224 3937 y(In)47 b(the)g(general)g(case,)53 b(the)47 b(isot)m(ypic)g(comp)s(onen)m(ts)g(can)h(ha)m(v)m(e)g(m)m(ultiplicit)m (y)c FD(m)3132 3951 y Fx(i;j)3265 3937 y FD(>)53 b FK(1.)92 b(If)125 4045 y FD(U)187 4060 y Fx(\014)227 4069 y FG(1)265 4045 y FD(;)15 b(:)g(:)g(:)i(;)e(U)529 4060 y Fx(\014)569 4072 y Fq(k)641 4045 y FK(are)31 b(the)g(isomorphic)d(comp)s(onen)m (ts,)j(then)f(the)g(condition)f(means)i(that)1212 4194 y FC(h)p FK(p)1298 4209 y Fx(\014)1338 4218 y FG(1)1376 4194 y FK(\()p FD(M)10 b FK(\))p FD(;)15 b(:)g(:)g(:)i(;)e FK(p)1797 4209 y Fx(\014)1837 4221 y Fq(k)1879 4194 y FK(\()p FD(M)10 b FK(\))p FC(i)2082 4208 y Fs(\003)2143 4194 y FC(\\)20 b FD(L)2286 4209 y Fx(\014)2358 4194 y FC(6)p FK(=)25 b FC(;)p FD(:)125 4344 y FK(Here)40 b FD(L)411 4359 y Fx(\014)498 4344 y FK(is)f(an)m(y)h(of)g(the)h FD(L)1122 4359 y Fx(\014)1162 4369 y Fq(j)1198 4344 y FK(,)h(since)e(all)f FD(Q)1706 4359 y Fx(\014)1746 4369 y Fq(j)1822 4344 y FK(are)h(the)g(same,)j(and)d FC(h)p FD(v)2680 4358 y FJ(1)2719 4344 y FD(;)15 b(:)g(:)g(:)i(;)e(v)2965 4358 y Fx(m)3032 4344 y FC(i)3067 4358 y Fs(\003)3147 4344 y FK(denotes)40 b(the)125 4452 y(collection)29 b(of)h(non-trivial) e(linear)h(com)m(binations)g(of)h FD(v)2031 4466 y FJ(1)2070 4452 y FD(;)15 b(:)g(:)g(:)i(;)e(v)2316 4466 y Fx(m)2383 4452 y FK(,)30 b(that)h(is)e(of)h(com)m(binations)f(of)h(the)125 4560 y(form)f FD(\025)392 4574 y FJ(1)432 4560 y FD(v)476 4574 y FJ(1)536 4560 y FK(+)20 b FD(:)15 b(:)g(:)21 b FK(+)f FD(\025)897 4574 y Fx(m)963 4560 y FD(v)1007 4574 y Fx(m)1104 4560 y FK(with)29 b(\()p FD(\025)1399 4574 y FJ(1)1439 4560 y FD(;)15 b(:)g(:)g(:)i(;)e(\025)1694 4574 y Fx(m)1761 4560 y FK(\))25 b FC(6)p FK(=)g(\(0)p FD(;)15 b(:)g(:)g(:)j(;)d FK(0\).)42 b(W)-8 b(e)31 b(de\014ne)f(an)g (algebraic)g(set)125 4787 y(\(21\))574 4787 y SDict begin H.S end 574 4787 a 574 4787 a SDict begin 13 H.A end 574 4787 a 574 4787 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.21) cvn /DEST pdfmark end 574 4787 a -77 x FC(L)25 b FK(=)758 4636 y Ft(\010)811 4710 y FD(M)g FC(j)15 b(h)p FK(p)1050 4725 y Fx(\014)1090 4734 y FG(1)1129 4710 y FK(\()p FD(M)10 b FK(\))p FD(;)15 b(:)g(:)g(:)i(;)e FK(p)1550 4725 y Fx(\014)1590 4737 y Fq(k)1632 4710 y FK(\()p FD(M)10 b FK(\))p FC(i)1835 4724 y Fs(\003)1896 4710 y FC(\\)19 b FD(L)2038 4725 y Fx(\014)2111 4710 y FC(6)p FK(=)25 b FC(;)p FD(;)979 4859 y FK(for)31 b(all)e FD(\027)1290 4873 y Fx(i)1343 4859 y FD(<)c(\014)1490 4873 y FJ(1)1530 4859 y FD(;)15 b(:)g(:)g(:)i(;)e(\014)1783 4874 y Fx(k)1851 4859 y FC(\024)25 b FD(\027)1992 4873 y Fx(i)p FJ(+1)2141 4859 y FK(isot)m(ypic)o FD(;)15 b(i)26 b FK(=)f(1)p FD(;)15 b(:)g(:)g(:)i(;)e(s)21 b FC(\000)f FK(1)3084 4785 y Ft(\011)3137 4859 y FD(:)125 5009 y FK(The)29 b(algebraicit)m(y)i(of)f(this)f (condition)g(is)h(equiv)-5 b(alen)m(t)30 b(to)h(the)f(non-v)-5 b(anishing)28 b(of)j(the)f(elemen)m(t)755 5192 y(p)806 5207 y Fx(\014)846 5216 y FG(1)884 5192 y FK(\()p FD(M)10 b FK(\))21 b FC(^)f FD(:)15 b(:)g(:)21 b FC(^)f FK(p)1412 5207 y Fx(\014)1452 5219 y Fq(k)1493 5192 y FK(\()p FD(M)10 b FK(\))26 b FC(2)1773 5105 y Ft(^)1874 5122 y Fx(k)1932 5192 y FK(\(Hom)16 b(\()p FD(U)2269 5207 y Fx(\014)2317 5192 y FD(;)f FK(Gr)2464 5206 y Fx(i)p Fs(\000)p FJ(1)2582 5192 y FK(\()p FD(\045)p FK(\)\))h FD(=L)2857 5207 y Fx(\014)2905 5192 y FK(\))f FD(:)224 5413 y FK(As)31 b(b)s(efore)e FC(M)740 5427 y Fx(\045)806 5413 y FK(=)c FC(H)c(\\)f(L)p FK(,)30 b(and)g FC(M)1485 5427 y Fx(\045)1551 5388 y FC(\030)1551 5417 y FK(=)1647 5413 y FC(M)1756 5428 y Fx(\045)1792 5409 y Fr(0)1849 5413 y FK(for)g(an)m(y)h FD(\045;)15 b(\045)2294 5380 y Fs(0)2342 5413 y FC(2)2436 5390 y FK(^)2428 5413 y FC(I)2478 5427 y FJ(0)2517 5413 y FK(\()p FD(\034)10 b FK(\).)879 b Fo(\003)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.15) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(15)p 0 TeXcolorgray 224 266 a FK(With)30 b(this)f(result)h(at)h(hand,)e(w)m(e)i(get)h(the)e (follo)m(wing.)125 431 y SDict begin H.S end 125 431 a 125 431 a SDict begin 13 H.A end 125 431 a 125 431 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.5) cvn /DEST pdfmark end 125 431 a 0 TeXcolorgray FL(Prop)s(osition)36 b(4.5.)p 0 TeXcolorgray 42 w Fp(Fix)d FD(\024)g Fp(and)g(a)g(typ)-5 b(e)34 b FD(\034)h FC(2)25 b(T)1798 445 y Fx(\024)1842 431 y Fp(.)42 b(The)33 b(map)1533 564 y FK(^)1503 587 y(Gr)1609 601 y Fs(\017)1674 587 y FK(:)1744 564 y(^)1725 587 y FD(R)p FK(\()p FD(\034)10 b FK(\))26 b FC(!)2064 564 y FK(^)2057 587 y FC(I)6 b FK(\()p FD(\034)k FK(\))125 739 y Fp(is)36 b(lo)-5 b(c)g(al)5 b(ly)38 b(trivial)f(in)f(the)g (Zariski)h(top)-5 b(olo)g(gy.)55 b(Mor)-5 b(e)g(over,)38 b(the)e(\014b)-5 b(er)37 b(of)f(this)h(map)h(is)e(the)g(quotient)125 846 y FK(\(\()p FC(M)304 860 y Fx(\045)365 846 y FC(\002)20 b FK(GL)584 860 y Fx(r)622 846 y FK(\))15 b FD(=)p FC(G)771 860 y Fx(\045)812 846 y FK(\))h FD(=)p FK(\()943 778 y Ft(Q)1029 873 y Fx(\013)1094 846 y FK(PGL)1284 861 y Fx(d)1320 869 y Fq(\013)1369 846 y FK(\))p Fp(,)33 b(wher)-5 b(e)33 b FD(d)1768 860 y Fx(\013)1843 846 y FK(=)25 b(dim)14 b FD(U)2168 860 y Fx(\013)2217 846 y Fp(.)p 0 TeXcolorgray 125 1061 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(Let)34 b(us)d(lo)s(ok)i(at)g(the)g(\014b)s(er)e (of)h(the)h(map)1872 1038 y(^)1841 1061 y(Gr)1948 1075 y Fs(\017)1988 1061 y FK(.)47 b(Fix)32 b FD(\045)d FC(2)2398 1038 y FK(^)2390 1061 y FC(I)7 b FK(\()p FD(\034)j FK(\))33 b(and)f(let)g FD(\045)2959 1075 y FJ(0)3028 1061 y FC(2)3125 1038 y FK(^)3117 1061 y FC(I)3167 1075 y FJ(0)3206 1061 y FK(\()p FD(\034)10 b FK(\))33 b(b)s(e)f(the)125 1169 y(diagonal)d(elemen)m(t)i(conjugated)g(to)g FD(\045)p FK(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)h(a)f(natural)e(map)841 1325 y(\011)c(:)h FC(M)1097 1339 y Fx(\045)1133 1348 y FG(0)1192 1325 y FC(\002)20 b FK(GL)1411 1339 y Fx(r)1474 1325 y FC(!)1610 1302 y FK(^)1590 1325 y FD(R)q FK(\()p FD(\034)10 b FK(\))p FD(;)107 b FK(\011\()p FD(M)5 b(;)15 b(Q)p FK(\))26 b(=)f FD(Q)p FK(\()p FD(\045)2534 1339 y FJ(0)2574 1325 y FK(\))2609 1339 y Fx(M)2688 1325 y FD(Q)2760 1287 y Fs(\000)p FJ(1)2869 1325 y FD(:)125 1479 y FK(Moreo)m(v)m(er,)40 b(\011\()p FD(M)5 b(;)15 b(Q)p FK(\))37 b(=)e(\011\()p FD(M)1250 1446 y Fs(0)1274 1479 y FD(;)15 b(Q)1386 1446 y Fs(0)1410 1479 y FK(\))37 b(if)e(and)h(only)g(if)g FD(Q)2122 1446 y Fs(0\000)p FJ(1)2235 1479 y FD(Q)p FK(\()p FD(\045)2389 1493 y FJ(0)2429 1479 y FK(\))2464 1493 y Fx(M)2543 1479 y FD(Q)2615 1446 y Fs(\000)p FJ(1)2710 1479 y FD(Q)2782 1446 y Fs(0)2841 1479 y FK(=)f(\()p FD(\045)3029 1493 y FJ(0)3069 1479 y FK(\))3104 1494 y Fx(M)3179 1476 y Fr(0)3206 1479 y FK(,)j(whic)m(h)e(in)125 1587 y(particular)28 b(means)j(that)f FD(Q)1091 1554 y Fs(0\000)p FJ(1)1205 1587 y FD(Q)25 b FC(2)g(G)1442 1601 y Fx(\045)1478 1610 y FG(0)1517 1587 y FK(.)224 1730 y(Hence,)36 b(if)d(w)m(e)h(quotien)m(t)g(b)m(y)g (the)g(natural)f(action)h(of)g(the)g(gauge)h(group)e(w)m(e)h(get)h (that)g(the)f(map)137 1836 y(~)125 1859 y(\011)g(:)h(\()p FC(M)434 1873 y Fx(\045)470 1882 y FG(0)529 1859 y FC(\002)20 b FK(GL)749 1873 y Fx(r)786 1859 y FK(\))c FD(=)p FC(G)936 1873 y Fx(\045)972 1882 y FG(0)1046 1859 y FC(!)1191 1836 y FK(^)1172 1859 y FD(R)p FK(\()p FD(\034)10 b FK(\))37 b(is)e(injectiv)m(e.)58 b(Its)36 b(image)g(is)2445 1836 y(^)2414 1859 y(Gr)2521 1799 y Fs(\000)p FJ(1)2521 1882 y Fs(\017)2615 1859 y FK(\()p FD(\045)2697 1873 y FJ(0)2737 1859 y FK(\))24 b FC(\002)2891 1791 y Ft(Q)2977 1886 y Fx(\013)3042 1859 y FK(PGL)3232 1874 y Fx(d)3268 1882 y Fq(\013)3316 1859 y FK(,)38 b(where)125 1967 y(the)29 b(second)g(factor)g(accoun)m(ts)i(for)d(mo)m(ving)h FD(\045)1702 1981 y FJ(0)1770 1967 y FK(in)f(its)g(isomorphism)e(class)j(\(recall)f (that)i(the)f(action)125 2075 y(of)h(PGL)418 2090 y Fx(d)454 2098 y Fq(\013)532 2075 y FK(on)g(eac)m(h)h(irreducible)26 b(piece)k(is)f(free\).)41 b(Quotien)m(ting)29 b(b)m(y)g(this)g(factor)i (the)f(iden)m(ti\014cation)125 2209 y(of)g(the)h(\014b)s(er)625 2186 y(^)594 2209 y(Gr)701 2149 y Fs(\000)p FJ(1)701 2231 y Fs(\017)796 2209 y FK(\()p FD(\045)878 2223 y FJ(0)917 2209 y FK(\))g(follo)m(ws.)40 b(The)30 b(Zariski)e(lo)s(cally) h(trivialit)m(y)f(comes)k(from)d(the)i(fact)g(that)g(the)125 2317 y(matrix)c(\()453 2317 y SDict begin H.S end 453 2317 a 0 TeXcolorgray FK(20)p 0 TeXcolorgray 544 2258 a SDict begin H.R end 544 2258 a 544 2317 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.20) cvn H.B /ANN pdfmark end 544 2317 a FK(\))h(can)g(b)s(e)f(de\014ned)g(in)f(a)j(Zariski)c(op)s (en)i(set,)j(whic)m(h)c(amoun)m(ts)i(to)h(diagonalizing)d FD(A)i FK(with)125 2425 y(a)i(basis)g(in)f(a)h(Zariski)f(neigh)m(b)s (orho)s(o)s(d.)2066 b Fo(\003)224 2626 y FK(Due)31 b(to)h(the)e (previous)g(result,)g(w)m(e)h(will)d(lo)s(osely)i(write)f(the)i (isomorphism)d(t)m(yp)s(e)j(of)g FC(M)3269 2640 y Fx(\045)3340 2626 y FK(and)f FC(G)3571 2640 y Fx(\045)125 2734 y FK(b)m(y)g FC(M)360 2748 y Fx(\034)433 2734 y FK(and)g FC(G)664 2748 y Fx(\034)707 2734 y FK(,)h(resp)s(ectiv)m(ely)-8 b(.)40 b(With)30 b(this)g(notation,)g(w)m(e)h(ha)m(v)m(e)h(that)125 2899 y SDict begin H.S end 125 2899 a 125 2899 a SDict begin 13 H.A end 125 2899 a 125 2899 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.6) cvn /DEST pdfmark end 125 2899 a 0 TeXcolorgray FL(Corollary)k(4.6.)p 0 TeXcolorgray 42 w Fp(F)-7 b(or)34 b(any)f(typ)-5 b(e)34 b FD(\034)h FC(2)25 b(T)1541 2913 y Fx(\024)1618 2899 y Fp(we)32 b(get)h(that)1141 3116 y FK([)1186 3094 y(^)1166 3116 y FD(R)q FK(\()p FD(\034)10 b FK(\)])26 b(=)1630 3055 y([)1663 3032 y(^)1655 3055 y FC(I)6 b FK(\()p FD(\034)k FK(\)])p 1513 3096 461 4 v 1513 3111 a Ft(Q)1599 3206 y Fx(\013)1649 3179 y FK([PGL)1864 3194 y Fx(d)1900 3202 y Fq(\013)1949 3179 y FK(])2004 3116 y FC(\001)2059 3055 y FK([)p FC(M)2193 3069 y Fx(\034)2237 3055 y FK(])20 b FC(\001)h FK([GL)2481 3069 y Fx(r)2519 3055 y FK(])p 2059 3096 486 4 v 2228 3179 a([)p FC(G)2307 3193 y Fx(\034)2351 3179 y FK(])2570 3116 y FD(:)p 0 TeXcolorgray 125 3370 a Fp(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(The)25 b(form)m(ula)f(follo)m(ws)h(directly)e(from)i(Prop)s(osition)2228 3370 y SDict begin H.S end 2228 3370 a 0 TeXcolorgray FK(4.5)p 0 TeXcolorgray 2344 3312 a SDict begin H.R end 2344 3312 a 2344 3370 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.5) cvn H.B /ANN pdfmark end 2344 3370 a 26 w FK(taking)g(in)m(to)g(accoun)m(t)h(that)g(virtual)125 3478 y(classes)k(are)h(m)m(ultiplicativ)m(e)d(for)i(Zariski)e(lo)s (cally)h(trivial)g(\014brations.)1002 b Fo(\003)224 3679 y FK(Observ)m(e)38 b(that,)j(writing)36 b(do)m(wn)i(the)g(m)m (ultiplicities)c(of)k(the)h(isot)m(ypic)e(comp)s(onen)m(ts)h(in)f(\() 3371 3679 y SDict begin H.S end 3371 3679 a 0 TeXcolorgray FK(13)p 0 TeXcolorgray 3462 3621 a SDict begin H.R end 3462 3621 a 3462 3679 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.13) cvn H.B /ANN pdfmark end 3462 3679 a FK(\))i(in)125 3787 y(terms)30 b(of)g(the)h(pieces)f FD(U)963 3801 y Fx(\013)1013 3787 y FK(,)g(w)m(e)h(ha)m(v)m(e)1224 3943 y([)1257 3920 y(^)1249 3943 y FC(I)7 b FK(\()p FD(\034)j FK(\)])p 1108 3984 461 4 v 1108 3999 a Ft(Q)1194 4094 y Fx(\013)1243 4067 y FK([PGL)1458 4082 y Fx(d)1494 4090 y Fq(\013)1543 4067 y FK(])1604 4005 y(=)1699 3918 y Ft(Y)1735 4109 y Fx(\013)1886 3864 y Ft(\002)1924 3937 y FD(R)1994 3904 y FJ(irr)1993 3960 y Fx(\024)2034 3968 y Fq(\013)2082 3864 y Ft(\003)p 1841 3984 326 4 v 1841 4067 a FK([PGL)2056 4082 y Fx(d)2092 4090 y Fq(\013)2141 4067 y FK(])2201 4005 y(=)2297 3918 y Ft(Y)2333 4109 y Fx(\013)2428 4005 y FB(M)2523 3967 y FJ(irr)2523 4027 y Fx(\024)2564 4035 y Fq(\013)2613 4005 y FD(;)125 4233 y FK(where)33 b FD(\024)443 4247 y Fx(\013)527 4233 y FK(denotes)i(the)g(con\014guration)e(of)i(eigen)m(v)-5 b(alues)34 b(of)g(the)h FD(\013)p FK(-blo)s(c)m(k.)52 b(W)-8 b(e)36 b(will)c(shorten)h(this)125 4341 y(last)d(factor)h(as)g FB(M)764 4308 y FJ(irr)764 4364 y Fx(\034)868 4341 y FK(=)964 4273 y Ft(Q)1050 4368 y Fx(\013)1115 4341 y FB(M)1210 4308 y FJ(irr)1210 4364 y Fx(\024)1251 4372 y Fq(\013)1299 4341 y FK(.)125 4507 y SDict begin H.S end 125 4507 a 125 4507 a SDict begin 13 H.A end 125 4507 a 125 4507 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.7) cvn /DEST pdfmark end 125 4507 a 0 TeXcolorgray FL(Corollary)g(4.7.)p 0 TeXcolorgray 39 w Fp(With)f(the)f(notation)i (of)e(\()1658 4507 y SDict begin H.S end 1658 4507 a 0 TeXcolorgray Fp(13)p 0 TeXcolorgray 1751 4448 a SDict begin H.R end 1751 4448 a 1751 4507 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.13) cvn H.B /ANN pdfmark end 1751 4507 a Fp(\),)h(if)e(for)i(every)f FD(m)2389 4521 y Fx(i;j)2494 4507 y FD(>)c FK(1)k Fp(we)g(have)h(that)g FK(dim)13 b FD(W)3435 4521 y Fx(i;j)3541 4507 y FK(=)125 4628 y(1)p Fp(,)32 b(i.e.)f(if)g(al)5 b(l)33 b(the)f(r)-5 b(ep)g(e)g(ate)g(d)34 b(irr)-5 b(e)g(ducible)33 b(r)-5 b(epr)g(esentations)35 b(ar)-5 b(e)32 b FK(1)p Fp(-dimensional,)i(then)e FC(I)7 b FK(\()p FD(\034)j FK(\))26 b(=)3442 4605 y(^)3435 4628 y FC(I)6 b FK(\()p FD(\034)k FK(\))125 4736 y Fp(and)33 b(the)g(map)1614 4850 y FD(R)q FK(\()p FD(\034)10 b FK(\))25 b FC(!)g(I)7 b FK(\()p FD(\034)j FK(\))125 4982 y Fp(is)32 b(lo)-5 b(c)g(al)5 b(ly)35 b(trivial)e(in)f(the)h(Zariski)g(top)-5 b(olo)g(gy)36 b(with)d(\014b)-5 b(er)33 b FK(\()q(\()p FC(M)2264 4996 y Fx(\045)2325 4982 y FC(\002)19 b FK(GL)2544 4996 y Fx(r)2582 4982 y FK(\))c FD(=)p FC(G)2731 4996 y Fx(\045)2772 4982 y FK(\))g FD(=)p FK(\()2902 4914 y Ft(Q)2989 5009 y Fx(\013)3054 4982 y FK(PGL)3244 4997 y Fx(d)3280 5005 y Fq(\013)3328 4982 y FK(\))p Fp(.)p 0 TeXcolorgray 125 5188 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(W)d(e)35 b(kno)m(w)f(that)g FD(R)1076 5155 y FJ(irr)1075 5211 y Fx(\024)1188 5188 y FK(is)f(a)h(single)e(p)s(oin)m (t)h(for)g(1-dimensional)f(represen)m(tations,)i(so)g(in)e(this)125 5305 y(case)h(the)f(action)h(of)f(the)h(symmetric)e(group)h FD(S)1776 5319 y Fx(m)1838 5329 y Fq(i;j)1948 5305 y FK(on)g(\()p FD(R)2181 5272 y FJ(irr)2180 5328 y Fx(\024)2260 5305 y FK(\))2295 5272 y Fx(m)2357 5282 y Fq(i;j)2468 5305 y FK(is)f(trivial.)44 b(Th)m(us,)3138 5282 y(^)3130 5305 y FC(I)6 b FK(\()p FD(\034)k FK(\))29 b(=)g FC(I)7 b FK(\()p FD(\034)j FK(\))125 5413 y(and)29 b(the)i(result)e(follo)m (ws)h(from)g(Prop)s(osition)1714 5413 y SDict begin H.S end 1714 5413 a 0 TeXcolorgray FK(4.5)p 0 TeXcolorgray 1830 5354 a SDict begin H.R end 1830 5354 a 1830 5413 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.5) cvn H.B /ANN pdfmark end 1830 5413 a FK(.)1686 b Fo(\003)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.16) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(16)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 224 266 a FK(Therefore,)38 b(under)d(the)i(h)m(yp)s(othesis)e(of)h(Corollary)2068 266 y SDict begin H.S end 2068 266 a 0 TeXcolorgray FK(4.7)p 0 TeXcolorgray 2184 207 a SDict begin H.R end 2184 207 a 2184 266 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.7) cvn H.B /ANN pdfmark end 2184 266 a FK(,)i(w)m(e)f(ha)m(v)m(e)h (the)e(analogous)h(form)m(ula)f(to)125 374 y(Corollary)527 374 y SDict begin H.S end 527 374 a 0 TeXcolorgray FK(4.6)p 0 TeXcolorgray 643 315 a SDict begin H.R end 643 315 a 643 374 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.6) cvn H.B /ANN pdfmark end 643 374 a 125 598 a FK(\(22\))1277 598 y SDict begin H.S end 1277 598 a 1277 598 a SDict begin 13 H.A end 1277 598 a 1277 598 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.22) cvn /DEST pdfmark end 1277 598 a FK([)p FD(R)q FK(\()p FD(\034)10 b FK(\)])26 b(=)f([)p FB(M)1759 561 y FJ(irr)1759 621 y Fx(\034)1838 598 y FK(])c FC(\001)1939 537 y FK([)p FC(M)2073 551 y Fx(\034)2117 537 y FK(])f FC(\001)h FK([GL)2361 551 y Fx(r)2399 537 y FK(])p 1939 577 486 4 v 2108 661 a([)p FC(G)2187 675 y Fx(\034)2230 661 y FK(])2434 598 y FD(:)224 859 y FK(In)27 b(particular,)g(for)h(rank)f FD(r)h FC(\024)d FK(4,)k(the)f(condition)f (of)h(Corollary)2437 859 y SDict begin H.S end 2437 859 a 0 TeXcolorgray FK(4.7)p 0 TeXcolorgray 2554 801 a SDict begin H.R end 2554 801 a 2554 859 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.7) cvn H.B /ANN pdfmark end 2554 859 a 27 w FK(holds.)39 b(The)27 b(only)g(exception)125 967 y(is)h FD(r)g FK(=)d(4,)30 b FD(s)25 b FK(=)g(1,)30 b FD(m)824 981 y FJ(1)p Fx(;)p FJ(1)944 967 y FK(=)25 b(2,)30 b(dim)14 b FD(W)1393 981 y FJ(1)p Fx(;)p FJ(1)1512 967 y FK(=)25 b(2,)30 b(but)f(in)f(this)g(case)j(all)d(represen)m (tations)i(are)g(reducible)125 1080 y(\(see)h(Section)625 1080 y SDict begin H.S end 625 1080 a 0 TeXcolorgray FK(8.2)p 0 TeXcolorgray 741 1021 a SDict begin H.R end 741 1021 a 741 1080 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.8.2) cvn H.B /ANN pdfmark end 741 1080 a FK(\),)g(so)g(it)f(do)s(es) g(not)g(con)m(tribute)g(to)i FD(R)2020 1047 y FJ(irr)2019 1102 y Fx(r)2098 1080 y FK(.)125 1268 y SDict begin H.S end 125 1268 a 125 1268 a SDict begin 13 H.A end 125 1268 a 125 1268 a SDict begin [/View [/XYZ H.V]/Dest (section.5) cvn /DEST pdfmark end 125 1268 a 996 1388 a FK(5.)46 b FE(Explicit)35 b(f)n(ormulas)d(f)n(or)h(some)h(stra)-6 b(t)g(a)224 1585 y FK(T)e(o)29 b(compute)f(the)h(motiv)m(e)g(of)f([)p FC(R)1376 1552 y FJ(irr)1455 1585 y FK(\()p FD($)s FK(\)])h(w)m(e)g (need)f(to)h(join)e(the)i(equations)f(\()2880 1585 y SDict begin H.S end 2880 1585 a 0 TeXcolorgray FK(10)p 0 TeXcolorgray 2971 1527 a SDict begin H.R end 2971 1527 a 2971 1585 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 2971 1585 a FK(\),)h(\()3095 1585 y SDict begin H.S end 3095 1585 a 0 TeXcolorgray FK(11)p 0 TeXcolorgray 3186 1527 a SDict begin H.R end 3186 1527 a 3186 1585 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.11) cvn H.B /ANN pdfmark end 3186 1585 a FK(\))g(and)f(\()3460 1585 y SDict begin H.S end 3460 1585 a 0 TeXcolorgray FK(22)p 0 TeXcolorgray 3551 1527 a SDict begin H.R end 3551 1527 a 3551 1585 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.22) cvn H.B /ANN pdfmark end 3551 1585 a FK(\),)125 1693 y(where)h FC(I)437 1707 y FJ(0)476 1693 y FK(\()p FD(\034)10 b FK(\),)32 b FC(M)762 1707 y Fx(\034)835 1693 y FK(and)e FC(G)1066 1707 y Fx(\034)1140 1693 y FK(are)g(giv)m(en)h(in)e(\()1670 1693 y SDict begin H.S end 1670 1693 a 0 TeXcolorgray FK(16)p 0 TeXcolorgray 1761 1635 a SDict begin H.R end 1761 1635 a 1761 1693 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.16) cvn H.B /ANN pdfmark end 1761 1693 a FK(\),)i(\()1887 1693 y SDict begin H.S end 1887 1693 a 0 TeXcolorgray FK(17)p 0 TeXcolorgray 1978 1635 a SDict begin H.R end 1978 1635 a 1978 1693 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.17) cvn H.B /ANN pdfmark end 1978 1693 a FK(\))g(and)f(\()2256 1693 y SDict begin H.S end 2256 1693 a 0 TeXcolorgray FK(18)p 0 TeXcolorgray 2347 1635 a SDict begin H.R end 2347 1635 a 2347 1693 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.18) cvn H.B /ANN pdfmark end 2347 1693 a FK(\).)224 1837 y(The)24 b(pro)s(of)g(of)h(Theorem)1113 1837 y SDict begin H.S end 1113 1837 a 0 TeXcolorgray FK(4.4)p 0 TeXcolorgray 1229 1778 a SDict begin H.R end 1229 1778 a 1229 1837 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.4) cvn H.B /ANN pdfmark end 1229 1837 a 25 w FK(is)f(constructiv)m(e)h (and)f(giv)m(es)h(explicit)e(descriptions)f(of)j FC(M)3250 1851 y Fx(\034)3318 1837 y FK(and)f FC(G)3543 1851 y Fx(\034)3586 1837 y FK(.)125 1945 y(Fix)34 b(t)m(w)m(o)i(v)-5 b(alues)34 b FD(\013;)15 b(\014)40 b FK(in)34 b(di\013eren)m(t)g(steps) h(of)f(the)h(semi-simple)d(\014ltration.)53 b(Let)35 b FD(\017)3034 1959 y FJ(1)3073 1945 y FD(;)15 b(:)g(:)g(:)i(;)e(\017) 3312 1959 y Fx(a)3388 1945 y FK(\(resp.)125 2066 y FD(")167 2080 y FJ(1)206 2066 y FD(;)g(:)g(:)g(:)i(;)e(")450 2081 y Fx(b)485 2066 y FK(\))32 b(b)s(e)e(the)i(eigen)m(v)-5 b(alues)31 b(of)g FD(A)1477 2080 y Fx(\013)1558 2066 y FK(and)g FD(A)1804 2081 y Fx(\014)1882 2066 y FK(\(resp.)g FD(B)2205 2080 y Fx(\013)2286 2066 y FK(and)f FD(B)2532 2081 y Fx(\014)2579 2066 y FK(\))i(and)f(let)g FD(a)3004 2033 y Fx(\013)3004 2094 y(k)3084 2066 y FK(and)g FD(a)3310 2022 y Fx(\014)3310 2095 y(k)3388 2066 y FK(\(resp.)125 2194 y FD(b)164 2161 y Fx(\013)164 2222 y(k)247 2194 y FK(and)i FD(b)466 2150 y Fx(\014)466 2223 y(k)513 2194 y FK(\))h(b)s(e)g(the)g(m)m(ultiplicities)29 b(of)34 b FD(\017)1560 2209 y Fx(k)1637 2194 y FK(in)e FD(A)1814 2208 y Fx(\013)1898 2194 y FK(and)h FD(A)2146 2209 y Fx(\014)2227 2194 y FK(\(resp.)h(of)g FD(")2633 2209 y Fx(k)2710 2194 y FK(in)e FD(B)2888 2208 y Fx(\013)2971 2194 y FK(and)i FD(B)3221 2209 y Fx(\014)3268 2194 y FK(\).)51 b(Then,)125 2322 y FD(b)164 2289 y Fx(\013)164 2350 y(k)213 2322 y FD(b)252 2278 y Fx(\014)252 2351 y(k)337 2322 y FK(is)36 b(the)i(n)m(um)m(b)s(er)e(of)i(times)f(that)h FD(")1544 2337 y Fx(k)1624 2322 y FK(is)f(a)h(rep)s(eated)g(eigen)m(v) -5 b(alue)37 b(in)f FD(B)2808 2336 y Fx(\013)2895 2322 y FK(and)h FD(B)3148 2337 y Fx(\014)3233 2322 y FK(and)g(th)m(us,)125 2450 y(according)c(to)h(Prop)s(osition)1144 2450 y SDict begin H.S end 1144 2450 a 0 TeXcolorgray FK(4.1)p 0 TeXcolorgray 1260 2392 a SDict begin H.R end 1260 2392 a 1260 2450 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.1) cvn H.B /ANN pdfmark end 1260 2450 a FK(,)g(the)g(elemen)m(ts)g(of)f FC(M)2069 2464 y FJ(1)2142 2450 y FK(ha)m(v)m(e)2355 2382 y Ft(P)2451 2477 y Fx(k)2509 2450 y FD(b)2548 2417 y Fx(\013)2548 2478 y(k)2597 2450 y FD(b)2636 2406 y Fx(\014)2636 2479 y(k)2717 2450 y FK(\014xed)g(elemen)m(ts.)50 b(So)33 b(the)125 2558 y(con)m(tribution)28 b(to)k(the)e(dimension)e (of)i FC(M)1552 2572 y FJ(1)1622 2558 y FK(of)h(the)f(pair)g FD(\013;)15 b(\014)36 b FK(is)1113 2749 y FD(C)7 b FK(\()p FD(\013;)15 b(\014)5 b FK(\))26 b(=)f(dim)14 b FD(U)1760 2763 y Fx(\013)1829 2749 y FC(\001)21 b FK(dim)14 b FD(U)2104 2764 y Fx(\014)2171 2749 y FC(\000)2262 2662 y Ft(X)2308 2860 y Fx(k)2408 2749 y FD(b)2447 2711 y Fx(\013)2447 2772 y(k)2497 2749 y FD(b)2536 2704 y Fx(\014)2536 2778 y(k)2598 2749 y FD(:)224 3050 y FK(De\014ne)30 b FD(d)p FK(\()p FD(\013)p FK(\))d(=)e FD(i)p FK(,)31 b(if)e FD(U)1035 3064 y Fx(\013)1114 3050 y FK(b)s(elongs)g(to)i(the)g FD(i)p FK(-th)f(step)g(of)g(the)h(semi-simple)c(\014ltration)i(i.e.)h FD(\027)3397 3064 y Fx(i)p Fs(\000)p FJ(1)3541 3050 y FD(<)125 3158 y(\013)j FC(\024)g FD(\027)365 3172 y Fx(i)393 3158 y FK(.)56 b(In)34 b(the)i(same)f(v)m(ein,)h(de\014ne)f FD(c)1516 3172 y Fx(i)1544 3158 y FK(\()p FD(\014)5 b FK(\))35 b(=)e FD(j)5 b FK(,)37 b(if)d FD(U)2063 3173 y Fx(\014)2146 3158 y FK(b)s(elongs)g(to)i(the)f FD(i)p FK(-th)h(step)f(of)g(the)h(semi-)125 3266 y(simple)24 b(\014ltration)i(and)h(it)f(is)g(a)h(comp)s(onen)m(t)h(of)f(the)g FD(j)5 b FK(-th)28 b(isot)m(ypic)e(factor)i FD(W)2775 3280 y Fx(i;j)2855 3266 y FK(,)g(that)g(is)e FD(U)3252 3281 y Fx(\014)3324 3241 y FC(\030)3324 3270 y FK(=)3420 3266 y FD(W)3506 3280 y Fx(i;j)3586 3266 y FK(.)125 3374 y(In)40 b(this)f(w)m(a)m(y)-8 b(,)45 b(the)c(con)m(tribution)f(to)h FC(M)1577 3388 y FJ(1)1658 3374 y FK(of)g(the)g(blo)s(c)m(ks)f(corresp) s(onding)f(to)i(the)g FD(j)5 b FK(-th)42 b(isot)m(ypic)125 3482 y(comp)s(onen)m(t)30 b(is)g(a)g(v)m(ector)i(space)f(of)g (dimension)1276 3672 y FD(C)1341 3686 y Fx(i;j)1446 3672 y FK(=)1584 3586 y Ft(X)1542 3787 y Fx(d)p FJ(\()p Fx(\013)p FJ(\)=)p Fx(i)1897 3586 y Ft(X)1803 3787 y Fx(c)1834 3797 y Fq(i)p FG(+1)1937 3787 y FJ(\()p Fx(\014)s FJ(\)=)p Fx(j)2137 3672 y FD(C)7 b FK(\()p FD(\013;)15 b(\014)5 b FK(\))p FD(:)224 3989 y FK(T)-8 b(o)30 b(this)d(con)m(tribution,)h(w) m(e)i(ha)m(v)m(e)g(to)g(subtract)f(the)g(elemen)m(ts)g(in)f FC(M)2622 4003 y FJ(1)2679 3989 y FC(\000)17 b(M)2876 4003 y Fx(\034)2948 3989 y FK(corresp)s(onding)27 b(to)125 4097 y(the)j(forbidden)e(con\014gurations.)40 b(These)30 b(are)h(giv)m(en)f(b)m(y)g(the)h(\\Sc)m(h)m(ub)s(ert)f(cells")g(\()2950 4097 y SDict begin H.S end 2950 4097 a 0 TeXcolorgray FK(21)p 0 TeXcolorgray 3041 4039 a SDict begin H.R end 3041 4039 a 3041 4097 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.21) cvn H.B /ANN pdfmark end 3041 4097 a FK(\),)418 4275 y FC(D)488 4289 y Fx(i;j)593 4275 y FK(=)689 4201 y Ft(\010)742 4275 y FC(h)p FK(p)828 4290 y Fx(\014)868 4299 y FG(1)907 4275 y FK(\()p FD(M)10 b FK(\))p FD(;)15 b(:)g(:)g(:)i(;)e FK(p)1328 4290 y Fx(\014)1368 4302 y Fq(k)1410 4275 y FK(\()p FD(M)10 b FK(\))p FC(i)1613 4289 y Fs(\003)1674 4275 y FC(\\)19 b FD(L)1816 4290 y Fx(\014)1889 4275 y FC(6)p FK(=)25 b FC(;)15 b(j)g FD(c)2124 4289 y Fx(i)p FJ(+1)2243 4275 y FK(\()p FD(\014)2329 4289 y FJ(1)2370 4275 y FK(\))25 b(=)g FD(:)15 b(:)g(:)26 b FK(=)f FD(c)2792 4289 y Fx(i)p FJ(+1)2911 4275 y FK(\()p FD(\014)2997 4290 y Fx(k)3041 4275 y FK(\))g(=)g FD(j)3239 4201 y Ft(\011)3293 4275 y FD(:)125 4452 y FK(Hence,)31 b(the)g(con)m(tribution)d(of)j(the)g(blo)s(c)m(ks)e(in)g(consecutiv)m (e)j(steps)e(is)1494 4586 y Fx(s)1452 4614 y Ft(Y)1453 4809 y Fx(i)p FJ(=1)1615 4584 y Fx(s)1648 4594 y Fq(i)1586 4614 y Ft(Y)1583 4809 y Fx(j)t FJ(=1)1721 4626 y Ft(\000)1762 4700 y FD(k)1812 4663 y Fx(C)1862 4673 y Fq(i;j)1961 4700 y FC(\000)20 b(D)2122 4714 y Fx(i;j)2202 4626 y Ft(\001)2259 4700 y FD(:)224 5007 y FK(On)34 b(the)h(other)g(hand,)g (the)g(con)m(tribution)e(of)i(the)g(non-consecutiv)m(e)h(blo)s(c)m(ks)e FD(\013;)15 b(\014)40 b FK(is)34 b(just)g(giv)m(en)125 5115 y(b)m(y)c(a)h(v)m(ector)h(space)e(of)h(dimension)1395 5306 y FD(C)h FK(=)1730 5219 y Ft(X)1588 5421 y Fx(d)p FJ(\()p Fx(\014)s FJ(\))p Fs(\000)p Fx(d)p FJ(\()p Fx(\013)p FJ(\))p Fx(>)p FJ(1)2019 5306 y FD(C)7 b FK(\()p FD(\013;)15 b(\014)5 b FK(\))p FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.17) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(17)p 0 TeXcolorgray 125 266 a FK(Therefore,)30 b(the)g(motiv)m(e)h(of)g FC(M)1229 280 y Fx(\034)1303 266 y FK(is)e(giv)m(en)h(b)m(y)125 483 y(\(23\))643 483 y SDict begin H.S end 643 483 a 643 483 a SDict begin 13 H.A end 643 483 a 643 483 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.23) cvn /DEST pdfmark end 643 483 a FK([)p FC(M)777 497 y Fx(\034)821 483 y FK(])25 b(=)g([)p FD(k)1042 445 y Fx(C)1102 483 y FK(])20 b FC(\001)1234 369 y Fx(s)1193 396 y Ft(Y)1194 592 y Fx(i)p FJ(=1)1356 367 y Fx(s)1389 377 y Fq(i)1327 396 y Ft(Y)1324 592 y Fx(j)t FJ(=1)1462 409 y Ft(\002)1499 483 y FD(k)1549 445 y Fx(C)1599 455 y Fq(i;j)1698 483 y FC(\000)g(D)1859 497 y Fx(i;j)1939 409 y Ft(\003)2002 483 y FK(=)25 b FD(q)2142 445 y Fx(C)2258 369 y(s)2216 396 y Ft(Y)2217 592 y Fx(i)p FJ(=1)2379 367 y Fx(s)2412 377 y Fq(i)2351 396 y Ft(Y)2348 592 y Fx(j)t FJ(=1)2485 409 y Ft(\000)2527 483 y FD(q)2571 445 y Fx(C)2621 455 y Fq(i;j)2719 483 y FC(\000)20 b FK([)p FC(D)2905 497 y Fx(i;j)2986 483 y FK(])3011 409 y Ft(\001)3068 483 y FD(:)224 764 y FK(No)m(w,)31 b(let)g(us)e(giv)m(e)i(form)m(ulas)f (for)g([)p FC(D)1495 778 y Fx(i;j)1575 764 y FK(])h(in)e(some)i (particular)e(cases.)125 844 y SDict begin H.S end 125 844 a 125 844 a SDict begin 13 H.A end 125 844 a 125 844 a SDict begin [/View [/XYZ H.V]/Dest (Item.1) cvn /DEST pdfmark end 125 844 a 0 TeXcolorgray 270 926 a FK(\(1\))p 0 TeXcolorgray 42 w(If)c FD(m)593 940 y Fx(i)p FJ(+1)p Fx(;j)789 926 y FK(=)g(1,)i(that)f(is)e(the)i(isot)m(ypic)f(comp)s (onen)m(t)h FD(W)2287 940 y Fx(i)p FJ(+1)p Fx(;j)2482 926 y FK(has)g(of)f(m)m(ultiplicit)m(y)e(1,)k(observ)m(e)427 1034 y(that)k(the)g(Sc)m(h)m(ub)s(ert)e(cell)h(is)f(just)689 1193 y FC(D)759 1207 y Fx(i;j)865 1193 y FK(=)c FD(L)1023 1208 y Fx(\014)1095 1193 y FK(=)1230 1107 y Ft(M)1191 1308 y Fx(d)p FJ(\()p Fx(\013)p FJ(\)=)p Fx(i)1422 1193 y FD(`)1460 1208 y Fx(\013\014)1577 1193 y FK(=)1712 1107 y Ft(M)1673 1308 y Fx(d)p FJ(\()p Fx(\013)p FJ(\)=)p Fx(i)1889 1193 y FC(h)p FD(Q)1996 1207 y Fx(\013)2045 1193 y FK(\002)p FD(Q)2188 1155 y Fs(\000)p FJ(1)2188 1222 y Fx(\014)2282 1193 y FD(B)2351 1208 y Fx(\014)2419 1193 y FC(\000)19 b FD(B)2578 1207 y Fx(\013)2628 1193 y FD(Q)2700 1207 y Fx(\013)2749 1193 y FK(\002)p FD(Q)2892 1155 y Fs(\000)p FJ(1)2892 1222 y Fx(\014)2986 1193 y FC(i)p FD(;)427 1448 y FK(where)k FD(\014)29 b FK(is)22 b(the)i(only)e(index)g(with)g FD(c)1666 1462 y Fx(i)p FJ(+1)1785 1448 y FK(\()p FD(\014)5 b FK(\))26 b(=)f FD(j)5 b FK(.)39 b(The)23 b(matrix)f(\002)h(mo)m(v)m(es)i(in)d(a)i(v)m (ector)h(space)427 1569 y(of)34 b(dimension)967 1501 y Ft(P)1063 1596 y Fx(k)1121 1569 y FD(a)1169 1536 y Fx(\013)1169 1597 y(k)1218 1569 y FD(a)1266 1525 y Fx(\014)1266 1599 y(k)1314 1569 y FK(.)50 b(The)33 b(k)m(ernel)g(of)h(the)g(map)f (\002)d FC(7!)h FD(Q)2617 1583 y Fx(\013)2666 1569 y FK(\002)p FD(Q)2809 1531 y Fs(\000)p FJ(1)2809 1599 y Fx(\014)2903 1569 y FD(B)2972 1584 y Fx(\014)3042 1569 y FC(\000)22 b FD(B)3204 1583 y Fx(\013)3253 1569 y FD(Q)3325 1583 y Fx(\013)3375 1569 y FK(\002)p FD(Q)3518 1531 y Fs(\000)p FJ(1)3518 1599 y Fx(\014)427 1688 y FK(is)41 b(giv)m(en)g(b)m(y)g(the)h(homomorphisms)d(as)i(represen)m(tations,)k (Hom)2753 1702 y FJ(\000)2801 1688 y FK(\()p FD(U)2898 1703 y Fx(\014)2946 1688 y FD(;)15 b(U)3048 1702 y Fx(\013)3098 1688 y FK(\),)45 b(since)40 b FD(S)49 b FK(=)427 1805 y FD(Q)499 1819 y Fx(\013)549 1805 y FK(\002)p FD(Q)692 1767 y Fs(\000)p FJ(1)692 1834 y Fx(\014)822 1805 y FK(is)35 b(a)i(homomorphism)d(of)i(represen)m(tations)h(if)e(and)g(only)h(if)f FD(S)5 b(B)3091 1820 y Fx(\014)3173 1805 y FK(=)35 b FD(B)3348 1819 y Fx(\013)3398 1805 y FD(S)41 b FK(\(as)427 1924 y FD(S)5 b(A)556 1939 y Fx(\014)630 1924 y FK(=)25 b FD(A)794 1938 y Fx(\013)844 1924 y FD(S)36 b FK(holds)29 b(b)m(y)i(the)g(condition)e(on)i(\002\).)42 b(Adding)29 b(up)h(o)m(v)m(er)i FD(\013)p FK(,)f(w)m(e)h(ha)m(v)m(e)g(that)f FC(D)3531 1938 y Fx(i;j)427 2032 y FK(is)f(subspace)f(of)i(dimension) 508 2191 y FD(D)583 2205 y Fx(i;j)688 2191 y FK(=)25 b(dim)14 b FD(L)1013 2206 y Fx(\014)1085 2191 y FK(=)1223 2105 y Ft(X)1181 2306 y Fx(d)p FJ(\()p Fx(\013)p FJ(\)=)p Fx(i)1412 2191 y FK(dim)f FD(`)1616 2206 y Fx(\013\014)1734 2191 y FK(=)1830 2105 y Ft(X)1876 2302 y Fx(k)1976 2191 y FD(a)2024 2153 y Fx(\013)2024 2214 y(k)2074 2191 y FD(a)2122 2147 y Fx(\014)2122 2220 y(k)2189 2191 y FC(\000)20 b FK(dim)14 b(Hom)2636 2205 y FJ(\000)2685 2191 y FK(\()p FD(U)2782 2206 y Fx(\014)2829 2191 y FD(;)h FK(Gr)2976 2205 y Fx(i)3005 2191 y FK(\()p FD(V)3093 2205 y Fs(\017)3132 2191 y FK(\)\))p FD(:)427 2446 y FK(Therefore,)31 b(the)f(form)m(ula)g (\()1390 2446 y SDict begin H.S end 1390 2446 a 0 TeXcolorgray FK(23)p 0 TeXcolorgray 1481 2387 a SDict begin H.R end 1481 2387 a 1481 2446 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.5.23) cvn H.B /ANN pdfmark end 1481 2446 a FK(\))h(can)f(b)s(e)g(written)g(as)125 2610 y(\(24\))742 2610 y SDict begin H.S end 742 2610 a 742 2610 a SDict begin 13 H.A end 742 2610 a 742 2610 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.24) cvn /DEST pdfmark end 742 2610 a FK([)p FC(M)876 2624 y Fx(\034)920 2610 y FK(])25 b(=)g([)p FD(k)1141 2573 y Fx(C)1201 2610 y FK(])20 b FC(\001)1347 2524 y Ft(Y)1292 2718 y Fx(m)1354 2728 y Fq(i;j)1428 2718 y Fx(>)p FJ(1)1533 2536 y Ft(\000)1575 2610 y FD(q)1619 2573 y Fx(C)1669 2583 y Fq(i;j)1767 2610 y FC(\000)g FK([)p FC(D)1953 2624 y Fx(i;j)2033 2610 y FK(])2058 2536 y Ft(\001)2121 2610 y FC(\001)2221 2524 y Ft(Y)2166 2718 y Fx(m)2228 2728 y Fq(i;j)2302 2718 y FJ(=1)2407 2536 y Ft(\000)2449 2610 y FD(q)2493 2573 y Fx(C)2543 2583 y Fq(i;j)2641 2610 y FC(\000)g FD(q)2776 2573 y Fx(D)2834 2583 y Fq(i;j)2912 2536 y Ft(\001)2968 2610 y FD(:)125 2785 y SDict begin H.S end 125 2785 a 125 2785 a SDict begin 13 H.A end 125 2785 a 125 2785 a SDict begin [/View [/XYZ H.V]/Dest (Item.2) cvn /DEST pdfmark end 125 2785 a 0 TeXcolorgray 270 2861 a FK(\(2\))p 0 TeXcolorgray 42 w(Supp)s(ose)36 b(that)i(dim)13 b FD(W)1247 2875 y Fx(i)p FJ(+1)p Fx(;j)1455 2861 y FK(=)36 b(1)i(and)f FD(m)1909 2875 y Fx(i)p FJ(+1)p Fx(;j)2116 2861 y FK(=)f(2.)62 b(Denote)39 b(b)m(y)e FD(\014)2858 2875 y FJ(1)2898 2861 y FD(;)15 b(\014)2989 2875 y FJ(2)3067 2861 y FK(the)37 b(indices)f(in)427 2969 y(the)c(\()p FD(i)22 b FK(+)f(1\)-th)32 b(step)g(with)e FD(c)1434 2983 y Fx(i)p FJ(+1)1553 2969 y FK(\()p FD(\014)1639 2983 y FJ(1)1679 2969 y FK(\))e(=)f FD(c)1879 2983 y Fx(i)p FJ(+1)1997 2969 y FK(\()p FD(\014)2083 2983 y FJ(2)2124 2969 y FK(\))g(=)g FD(j)5 b FK(.)45 b(Supp)s(ose)30 b(also)i(that)g FD(s)3179 2983 y Fx(i)3234 2969 y FK(=)27 b(1,)33 b(with)427 3077 y FD(\013)42 b FK(=)e FD(\027)683 3091 y Fx(i)751 3077 y FK(the)g(only)e(index)h(in)f(the)i FD(i)p FK(-th)g(step.)69 b(In)39 b(that)h(case,)j(p)2713 3092 y Fx(\014)2753 3101 y FG(1)2791 3077 y FK(\()p FD(M)10 b FK(\))p FD(;)15 b FK(p)3050 3092 y Fx(\014)3090 3101 y FG(2)3129 3077 y FK(\()p FD(M)10 b FK(\))42 b FC(2)e FD(k)j FK(so)427 3185 y(alw)m(a)m(ys)29 b(0)d FC(2)f(h)p FK(p)959 3200 y Fx(\014)999 3209 y FG(1)1037 3185 y FK(\()p FD(M)10 b FK(\))p FD(;)15 b FK(p)1296 3200 y Fx(\014)1336 3209 y FG(2)1375 3185 y FK(\()p FD(M)10 b FK(\))p FC(i)1578 3199 y Fs(\003)1635 3185 y FC(\\)16 b FD(L)1774 3200 y Fx(\014)1821 3185 y FK(.)40 b(In)28 b(other)g(w)m(orks,)i(b)m(y)e(a)h (c)m(hange)g(of)g(basis,)f(on)g(the)427 3293 y(v)m(ector)34 b(\(p)788 3308 y Fx(\014)828 3317 y FG(1)866 3293 y FK(\()p FD(M)10 b FK(\))p FD(;)15 b FK(p)1125 3308 y Fx(\014)1165 3317 y FG(2)1204 3293 y FK(\()p FD(M)10 b FK(\)\))33 b(w)m(e)g(can)f(alw)m(a)m(ys)g(arrange)h(that)f(a)h(comp)s(onen)m(t)f (is)f(zero.)46 b(This)427 3401 y(implies)28 b(that)j FD(D)1008 3415 y Fx(i;j)1114 3401 y FK(=)24 b FD(C)1274 3415 y Fx(i;j)1355 3401 y FK(,)30 b(and)g(hence)g([)p FC(M)1973 3415 y Fx(\034)2017 3401 y FK(])c(=)e(0.)42 b(So)30 b(this)f(case)j(cannot)f(happ)s(en.)125 3427 y SDict begin H.S end 125 3427 a 125 3427 a SDict begin 13 H.A end 125 3427 a 125 3427 a SDict begin [/View [/XYZ H.V]/Dest (Item.3) cvn /DEST pdfmark end 125 3427 a 0 TeXcolorgray 270 3509 a FK(\(3\))p 0 TeXcolorgray 42 w(Supp)s(ose)j(that)j(dim)14 b FD(W)1245 3523 y Fx(i)p FJ(+1)p Fx(;j)1449 3509 y FK(=)35 b(1)h(and)g FD(m)1899 3523 y Fx(i)p FJ(+1)p Fx(;j)2103 3509 y FK(=)f(2)h(but)f(no)m(w)h FD(s)2700 3523 y Fx(i)2763 3509 y FK(=)e(2,)k(with)d(t)m(w)m(o)i(blo)s (c)m(ks)427 3617 y FD(U)489 3631 y Fx(\013)534 3640 y FG(1)573 3617 y FD(;)15 b(U)675 3631 y Fx(\013)720 3640 y FG(2)781 3617 y FK(of)22 b(m)m(ultiplicit)m(y)d(1.)38 b(Denote)23 b(the)f(eigen)m(v)-5 b(alues)21 b(of)h FD(A)2538 3631 y Fx(\013)2583 3640 y FG(1)2622 3617 y FD(;)15 b(A)2730 3631 y Fx(\013)2775 3640 y FG(2)2836 3617 y FK(\(resp.)22 b FD(B)3150 3631 y Fx(\013)3195 3640 y FG(1)3255 3617 y FK(and)f FD(B)3492 3631 y Fx(\013)3537 3640 y FG(2)3576 3617 y FK(\))427 3725 y(b)m(y)35 b FD(\017)595 3739 y Fx(\013;)p FJ(1)699 3725 y FD(;)15 b(\017)776 3739 y Fx(\013;)p FJ(2)916 3725 y FK(\(resp.)34 b FD(")1215 3739 y Fx(\013;)p FJ(1)1320 3725 y FD(;)15 b(")1402 3739 y Fx(\013;)p FJ(2)1507 3725 y FK(\))35 b(and)f(let)h(the)g(eigen)m(v)m (ector)i(of)e FD(A)2708 3740 y Fx(\014)2748 3749 y FG(1)2819 3725 y FK(=)d FD(A)2990 3740 y Fx(\014)3030 3749 y FG(2)3103 3725 y FK(\(resp.)j FD(B)3430 3740 y Fx(\014)3470 3749 y FG(1)3541 3725 y FK(=)427 3833 y FD(B)496 3848 y Fx(\014)536 3857 y FG(2)575 3833 y FK(\))g(b)s(e)g FD(\017)811 3848 y Fx(\014)894 3833 y FK(\(resp.)g FD(")1194 3848 y Fx(\014)1241 3833 y FK(\).)57 b(Supp)s(ose)33 b(that)j FD(\017)1958 3847 y Fx(\013;)p FJ(1)2063 3833 y FD(;)15 b(\017)2140 3847 y Fx(\013;)p FJ(2)2278 3833 y FC(6)p FK(=)33 b FD(\017)2419 3848 y Fx(\014)2502 3833 y FK(and)i FD(")2726 3847 y Fx(\013;)p FJ(1)2830 3833 y FD(;)15 b(")2912 3847 y Fx(\013;)p FJ(2)3051 3833 y FC(6)p FK(=)34 b FD(")3198 3848 y Fx(\014)3245 3833 y FK(.)56 b(In)35 b(that)427 3941 y(case,)d(w)m(e)e(ha)m(v)m(e)i (that)f(\002)25 b(=)g(0)30 b(and)g(th)m(us)f FD(`)1866 3956 y Fx(\013)1911 3965 y FG(1)1946 3956 y Fx(\014)2019 3941 y FK(=)c FD(`)2153 3956 y Fx(\013)2198 3965 y FG(2)2232 3956 y Fx(\014)2305 3941 y FK(=)g(0.)41 b(This)28 b(implies)g(that)i FD(L)3286 3956 y Fx(\014)3359 3941 y FK(=)25 b(0)30 b(so)427 4048 y(the)23 b(Sc)m(h)m(ub)s(ert)f(cell)g(condition)f(is)h(that)i(0)h FC(2)g(h)p FK(p)2011 4063 y Fx(\014)2051 4072 y FG(1)2089 4048 y FK(\()p FD(M)10 b FK(\))p FD(;)15 b FK(p)2348 4063 y Fx(\014)2388 4072 y FG(2)2427 4048 y FK(\()p FD(M)10 b FK(\))p FC(i)2630 4062 y Fs(\003)2671 4048 y FK(,)24 b(where)f(p)3027 4063 y Fx(\014)3067 4072 y FG(1)3105 4048 y FK(\()p FD(M)10 b FK(\))p FD(;)15 b FK(p)3364 4063 y Fx(\014)3404 4072 y FG(2)3443 4048 y FK(\()p FD(M)10 b FK(\))427 4156 y(are)35 b(t)m(w)m(o)g(2-dimensional)d(v)m(ectors.)52 b(This)32 b(happ)s(ens)g(if)h(and)h(only)f(if)f(p)2871 4171 y Fx(\014)2911 4180 y FG(1)2949 4156 y FK(\()p FD(M)10 b FK(\))p FD(;)15 b FK(p)3208 4171 y Fx(\014)3248 4180 y FG(2)3287 4156 y FK(\()p FD(M)10 b FK(\))35 b(are)427 4264 y(linearly)d(dep)s(enden)m(t,)i(so)h(the)f(corresp)s(onding)e(Sc)m (h)m(ub)s(ert)h(cell)h(are)g(the)h(matrices)f(of)g(rank)427 4372 y FC(\024)25 b FK(1.)41 b(That)31 b(is,)e([)p FC(D)1073 4386 y Fx(i;j)1154 4372 y FK(])c(=)g FD(q)1344 4339 y FJ(4)1404 4372 y FC(\000)20 b FK([GL)1648 4386 y FJ(2)1688 4372 y FK(])25 b(=)g FD(q)1878 4339 y FJ(3)1937 4372 y FK(+)20 b FD(q)2072 4339 y FJ(2)2132 4372 y FC(\000)g FD(q)s FK(.)125 4530 y SDict begin H.S end 125 4530 a 125 4530 a SDict begin 13 H.A end 125 4530 a 125 4530 a SDict begin [/View [/XYZ H.V]/Dest (theorem.5.1) cvn /DEST pdfmark end 125 4530 a 0 TeXcolorgray FL(Remark)43 b(5.1.)p 0 TeXcolorgray 46 w FK(These)37 b(are)i(the)f(only)f(cases)i(that)f(w)m (e)g(will)e(\014nd)g(when)h(computing)g(irreducible)125 4638 y(represen)m(tations)30 b(for)g FD(r)e FK(=)d(4.)224 4781 y(Observ)m(e)j(that)h(the)g(cases)g FD(\017)1184 4795 y Fx(\013;)p FJ(1)1314 4781 y FK(=)c FD(\017)1447 4796 y Fx(\014)1522 4781 y FK(or)k FD(")1674 4795 y Fx(\013;)p FJ(1)1804 4781 y FK(=)c FD(")1942 4796 y Fx(\014)2018 4781 y FK(in)i(item)h(\()2362 4781 y SDict begin H.S end 2362 4781 a 0 TeXcolorgray FK(3)p 0 TeXcolorgray 2408 4723 a SDict begin H.R end 2408 4723 a 2408 4781 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.3) cvn H.B /ANN pdfmark end 2408 4781 a FK(\))g(do)h(not)f(need)h(to)g(b)s(e)f(considered)125 4889 y(since)f(in)f(that)j(case)g(there)f(is)f(triple)f(eigen)m(v)m (ector)k(in)d(either)g FD(A)h FK(or)g FD(B)33 b FK(and)27 b(a)h(double)f(eigen)m(v)m(ector)j(in)125 4997 y(the)g(other)h(matrix,) f(so)g(all)g(the)g(represen)m(tations)g(are)h(reducible)d(\(c.f.)k (Remark)2932 4997 y SDict begin H.S end 2932 4997 a 0 TeXcolorgray FK(3.5)p 0 TeXcolorgray 3048 4939 a SDict begin H.R end 3048 4939 a 3048 4997 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.5) cvn H.B /ANN pdfmark end 3048 4997 a FK(\).)224 5191 y(A)h(form)m(ula)g(for)g FC(G)858 5205 y Fx(\034)934 5191 y FK(is)f(easier)h(to)h(obtain.)48 b(Recall)33 b(the)g(de\014nition)e(\()2602 5191 y SDict begin H.S end 2602 5191 a 0 TeXcolorgray FK(18)p 0 TeXcolorgray 2693 5132 a SDict begin H.R end 2693 5132 a 2693 5191 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.18) cvn H.B /ANN pdfmark end 2693 5191 a FK(\).)49 b(An)m(y)33 b FD(P)43 b FC(2)30 b(G)3245 5205 y Fx(\045)3318 5191 y FK(induces)125 5299 y(an)e(automorphism)f(of)h(Gr)1057 5313 y Fx(i)1086 5299 y FK(\()p FD(\045)p FK(\),)h(resp)s(ecting)f(the) h(comp)s(onen)m(ts.)40 b(By)29 b(Sc)m(h)m(ur)f(lemma,)g(it)g(acts)i(on) e(the)125 5406 y(isot)m(ypic)37 b(comp)s(onen)m(ts,)j(so)e(pro)s (ducing)1555 5338 y Ft(Q)1641 5365 y Fx(s)1641 5433 y(i)p FJ(=1)1774 5338 y Ft(Q)1860 5365 y Fx(s)1893 5375 y Fq(i)1860 5433 y Fx(j)t FJ(=1)2002 5406 y FK(GL)2130 5420 y Fx(m)2192 5430 y Fq(i;j)2270 5406 y FK(.)63 b(This)36 b(has)i(to)h(b)s(e)e (completed)h(with)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.18) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(18)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 125 266 a FK(upp)s(er)39 b(diagonal)h(blo)s(c)m(ks.)73 b(The)40 b(matrix)h FD(P)54 b FK(m)m(ust)41 b(\014x)g(the)g(diagonal)g(matrix)f FD(A)p FK(,)45 b(so)c(the)g(upp)s(er)125 374 y(diagonal)33 b(blo)s(c)m(k)g(of)h FD(P)47 b FK(corresp)s(onding)32 b(to)i FD(\013;)15 b(\014)40 b FK(has)34 b(to)h(b)s(e)e FD(S)j FK(=)31 b FD(Q)2512 388 y Fx(\013)2561 374 y FK(\002)p FD(Q)2704 335 y Fs(\000)p FJ(1)2704 403 y Fx(\014)2798 374 y FK(,)k(as)f(in)e(Theorem)3470 374 y SDict begin H.S end 3470 374 a 0 TeXcolorgray FK(4.4)p 0 TeXcolorgray 3586 315 a SDict begin H.R end 3586 315 a 3586 374 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.4) cvn H.B /ANN pdfmark end 3586 374 a FK(.)125 515 y(The)25 b(matrix)g(\002)g (has)851 446 y Ft(P)947 541 y Fx(k)1005 515 y FD(a)1053 482 y Fx(\013)1053 542 y(k)1103 515 y FD(a)1151 470 y Fx(\014)1151 544 y(k)1224 515 y FK(free)g(en)m(tries,)i(so)f(adding)e (up)h(on)h(the)f(p)s(ossible)f(matrices,)j(the)e(upp)s(er)125 622 y(diagonal)k(parts)h(form)g(a)h(v)m(ector)h(space)f(of)f(dimension) 1408 802 y FD(D)e FK(=)1704 716 y Ft(X)1607 917 y Fx(d)p FJ(\()p Fx(\014)s FJ(\))p Fx(>d)p FJ(\()p Fx(\013)p FJ(\))1948 716 y Ft(X)1994 913 y Fx(k)2095 802 y FD(a)2143 765 y Fx(\013)2143 825 y(k)2192 802 y FD(a)2240 758 y Fx(\014)2240 831 y(k)2302 802 y FD(:)125 1071 y FK(Hence)125 1291 y(\(25\))1373 1291 y SDict begin H.S end 1373 1291 a 1373 1291 a SDict begin 13 H.A end 1373 1291 a 1373 1291 a SDict begin [/View [/XYZ H.V]/Dest (equation.5.25) cvn /DEST pdfmark end 1373 1291 a FK([)p FC(G)1452 1305 y Fx(\034)1496 1291 y FK(])d(=)g FD(q)1686 1254 y Fx(D)1807 1178 y(s)1765 1205 y Ft(Y)1766 1400 y Fx(i)p FJ(=1)1928 1176 y Fx(s)1961 1186 y Fq(i)1899 1205 y Ft(Y)1896 1400 y Fx(j)t FJ(=1)2019 1291 y FK([GL)2172 1305 y Fx(m)2234 1315 y Fq(i;j)2312 1291 y FK(])p FD(:)125 1630 y SDict begin H.S end 125 1630 a 125 1630 a SDict begin 13 H.A end 125 1630 a 125 1630 a SDict begin [/View [/XYZ H.V]/Dest (section*.1) cvn /DEST pdfmark end 125 1630 a FL(Multiplicit)m(y.)46 b FK(Recall)35 b(that,)i(giv)m(en)e(a)h(m)m(ultiset)f FD(X)41 b FK(=)33 b FC(f)p FD(x)2244 1590 y Fx(m)2306 1599 y FG(1)2244 1656 y FJ(1)2345 1630 y FD(;)15 b(:)g(:)g(:)i(;)e(x)2599 1597 y Fx(m)2661 1605 y Fq(n)2599 1653 y Fx(n)2708 1630 y FC(g)q FK(,)37 b(w)m(e)e(can)h(consider)e(the)125 1738 y(subgroup)26 b FD(S)573 1752 y Fx(X)665 1738 y FC(\032)f FD(S)817 1752 y Fx(n)892 1738 y FK(of)j(p)s(erm)m(utations)e FD(\031)31 b FK(that)e(preserv)m(e)f(the)g(m)m(ultiplicit)m(y)d(of)j (the)g(m)m(ultiset,)g(that)125 1846 y(is)h FD(m)296 1864 y Fx(\031)r FJ(\()p Fx(i)p FJ(\))447 1846 y FK(=)c FD(m)623 1860 y Fx(i)681 1846 y FK(for)30 b(all)g(1)25 b FC(\024)g FD(i)h FC(\024)f FD(n)p FK(.)224 1993 y(No)m(w,)31 b(consider)d(a)i (con\014guration)f(of)g(eigen)m(v)-5 b(alues)30 b FD(\024)25 b FK(=)g(\()p Fn(\017)p FD(;)15 b Fn(")p FK(\))31 b(and)e(let)g FD(S)2774 2007 y Fx(\024)2844 1993 y FK(=)c FD(S)2996 2007 y Fm(\017)3052 1993 y FC(\002)18 b FD(S)3197 2007 y Fm(")3238 1993 y FK(.)41 b(Giv)m(en)29 b(a)125 2101 y(t)m(yp)s(e)j FD(\034)37 b FK(=)28 b(\()p FD(\030)t(;)15 b(\033)s FK(\))29 b FC(2)e(T)883 2115 y Fx(\024)959 2101 y FK(and)32 b FD(\031)e FC(2)e FD(S)1365 2115 y Fx(\024)1441 2101 y FK(w)m(e)k(set)h FD(\031)24 b FC(\001)d FD(\034)38 b FK(=)28 b(\()p FD(\030)t(;)15 b(\031)25 b FC(\001)c FD(\033)s FK(\).)46 b(That)32 b(is,)f(the)h(new)g(t)m(yp)s(e)g(has)f (the)125 2209 y(same)c(shap)s(e)f FD(\030)31 b FK(but)c(the)g(new)g (eigen)m(v)-5 b(alues,)28 b FD(\031)16 b FC(\001)e FD(\033)s FK(,)28 b(are)g(the)f(ones)h(of)f FD(\033)j FK(p)s(erm)m(uted)c (according)h(to)h(the)125 2317 y(p)s(erm)m(utation)h FD(\031)s FK(.)41 b(This)28 b(induces)h(an)h(action)h(of)g FD(S)1856 2331 y Fx(\024)1930 2317 y FK(on)g FC(T)2107 2331 y Fx(\024)2151 2317 y FK(.)224 2460 y(Moreo)m(v)m(er,)j(b)m(y)d (the)h(computations)f(ab)s(o)m(v)m(e,)i(w)m(e)e(ha)m(v)m(e)i(that)f([)p FD(R)q FK(\()p FD(\034)10 b FK(\)])27 b(=)g([)p FD(R)q FK(\()p FD(\031)d FC(\001)d FD(\034)10 b FK(\)].)44 b(This)29 b(implies)125 2568 y(that,)38 b(if)c(w)m(e)j(denote)f(the)g(length)f (of)h(the)g(orbit)f(of)h FD(\034)46 b FK(b)m(y)36 b FD(m)2231 2582 y Fx(\024)2275 2568 y FK(\()p FD(\034)10 b FK(\),)38 b(called)d(the)h(m)m(ultiplicit)m(y)d(of)j(the)125 2676 y(t)m(yp)s(e)30 b FD(\034)10 b FK(,)31 b(w)m(e)g(can)f(write)1196 2731 y Ft(X)1179 2928 y Fx(\034)8 b Fs(2T)1304 2936 y Fq(\024)1345 2818 y FK([)p FD(R)q FK(\()p FD(\034)i FK(\)])26 b(=)1803 2731 y Ft(X)1707 2933 y FJ([)p Fx(\034)8 b FJ(])p Fs(2T)1872 2941 y Fq(\024)1911 2933 y Fx(=S)1989 2941 y Fq(\024)2045 2818 y FD(m)2125 2832 y Fx(\024)2170 2818 y FK(\()p FD(\034)i FK(\)[)p FD(R)q FK(\()p FD(\034)g FK(\)])p FD(:)125 3069 y FK(This)30 b(form)m(ula)h(reduces)g (drastically)f(the)i(n)m(um)m(b)s(er)f(of)h(t)m(yp)s(es)g(that)g(need)g (to)h(b)s(e)e(considered)g(and)g(it)125 3177 y(is)e(the)i(v)m(ersion)e (that)i(w)m(e)g(will)d(use)i(in)f(Sections)1801 3177 y SDict begin H.S end 1801 3177 a 0 TeXcolorgray FK(7.1)p 0 TeXcolorgray 1917 3119 a SDict begin H.R end 1917 3119 a 1917 3177 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.7.1) cvn H.B /ANN pdfmark end 1917 3177 a 31 w FK(and)2124 3177 y SDict begin H.S end 2124 3177 a 0 TeXcolorgray FK(8)p 0 TeXcolorgray 2170 3119 a SDict begin H.R end 2170 3119 a 2170 3177 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.8) cvn H.B /ANN pdfmark end 2170 3177 a FK(.)125 3311 y SDict begin H.S end 125 3311 a 125 3311 a SDict begin 13 H.A end 125 3311 a 125 3311 a SDict begin [/View [/XYZ H.V]/Dest (section.6) cvn /DEST pdfmark end 125 3311 a 1302 3455 a FK(6.)47 b FE(Count)33 b(of)h(components)224 3652 y FK(In)43 b(this)f(section,)47 b(w)m(e)d(are)g(going)f(to)h(coun)m(t)g(the)g(v)-5 b(alid)41 b(con\014gurations)i(of)g(eigen)m(v)-5 b(alues)43 b FD(\024)48 b FK(=)125 3760 y(\()p FD(\024)212 3774 y FJ(1)252 3760 y FD(;)15 b(\024)344 3774 y FJ(2)384 3760 y FK(\).)47 b(As)32 b(w)m(e)h(will)d(see,)k(w)m(e)e(need)h(to)g(consider)e (di\013eren)m(t)h(cases)h(according)f(to)i(the)e(n)m(um)m(b)s(er)f(of) 125 3868 y(rep)s(eated)e(eigen)m(v)-5 b(alues)29 b(in)f FD(\024)1121 3882 y FJ(1)1191 3868 y FK(and)g FD(\024)1418 3882 y FJ(2)1458 3868 y FK(.)41 b(W)-8 b(e)30 b(will)d(analyze)j (completely)f(the)h(patterns)f(that)h(app)s(ear)125 3976 y(for)g FD(r)d FK(=)e(4,)31 b(and)f(the)h(general)f(case)i(when)d FD(r)m(;)15 b(n;)g(m)31 b FK(are)g(coprime)e(is)h(studied)e(in)i (Section)3220 3976 y SDict begin H.S end 3220 3976 a 0 TeXcolorgray FK(6.5)p 0 TeXcolorgray 3336 3917 a SDict begin H.R end 3336 3917 a 3336 3976 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.6.5) cvn H.B /ANN pdfmark end 3336 3976 a FK(.)224 4119 y(W)-8 b(e)42 b(\014x)e FD(n;)15 b(m)41 b FK(and)f(the)h(torus)g(knot)g(group)f(\()1867 4119 y SDict begin H.S end 1867 4119 a 0 TeXcolorgray FK(5)p 0 TeXcolorgray 1913 4060 a SDict begin H.R end 1913 4060 a 1913 4119 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.5) cvn H.B /ANN pdfmark end 1913 4119 a FK(\).)73 b(Fix)40 b(also)h FD(r)k(>)e FK(0)e(and)f(recall)g(that)i(w)m(e)f(are) 125 4228 y(in)m(terested)33 b(in)f(the)h(v)-5 b(ariet)m(y)34 b(\()1152 4228 y SDict begin H.S end 1152 4228 a 0 TeXcolorgray FK(6)p 0 TeXcolorgray 1198 4169 a SDict begin H.R end 1198 4169 a 1198 4228 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.6) cvn H.B /ANN pdfmark end 1198 4228 a FK(\),)g(and)f(particularly)-8 b(,)32 b(w)m(e)i(w)m(an)m(t)g(to) g(compute)f FC(R)2910 4195 y FJ(irr)2910 4250 y Fx(r)3022 4228 y FK(using)f(\()3298 4228 y SDict begin H.S end 3298 4228 a 0 TeXcolorgray FK(10)p 0 TeXcolorgray 3389 4169 a SDict begin H.R end 3389 4169 a 3389 4228 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 3389 4228 a FK(\).)50 b(By)125 4335 y(Lemma)453 4335 y SDict begin H.S end 453 4335 a 0 TeXcolorgray FK(3.1)p 0 TeXcolorgray 569 4277 a SDict begin H.R end 569 4277 a 569 4335 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.1) cvn H.B /ANN pdfmark end 569 4335 a 34 w FK(these)35 b(are)g(divided)c(according)j(to)h(a)g(ro)s(ot)g(of)f(unit)m(y)f FD($)i FC(2)c FD(\026)2732 4349 y Fx(r)2770 4335 y FK(.)52 b(If)34 b FD(\017)2979 4349 y FJ(1)3018 4335 y FD(;)15 b(:)g(:)g(:)i(;)e(\017)3257 4349 y Fx(r)3330 4335 y FK(are)34 b(the)125 4443 y(eigen)m(v)-5 b(alues)35 b(of)g FD(A)p FK(,)i(then)d FD(\017)1086 4410 y Fx(n)1086 4469 y(i)1167 4443 y FK(=)e FD($)38 b FK(for)d(all)f FD(i)p FK(,)j(and)e FD(\017)1970 4457 y FJ(1)2024 4443 y FC(\001)15 b(\001)g(\001)h FD(\017)2182 4457 y Fx(r)2254 4443 y FK(=)32 b(1.)56 b(T)-8 b(o)36 b(coun)m(t)f(the)h(distribution)31 b(of)125 4551 y(eigen)m(v)-5 b(alues,)30 b(w)m(e)h(in)m(tro)s(duce)e(the)i(set) 125 4722 y(\(26\))829 4722 y SDict begin H.S end 829 4722 a 829 4722 a SDict begin 13 H.A end 829 4722 a 829 4722 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.26) cvn /DEST pdfmark end 829 4722 a 855 4699 a FK(^)829 4722 y FD(N)902 4736 y Fx(n;r)1002 4722 y FK(\()p FD($)s FK(\))26 b(=)1272 4648 y Ft(\010)1325 4722 y FK(\()p FD(\017)1397 4736 y FJ(1)1437 4722 y FD(;)15 b(:)g(:)g(:)h(;)f(\017)1675 4736 y Fx(r)1714 4722 y FK(\))25 b FC(2)g FD(\026)1915 4684 y Fx(r)1915 4744 y(nr)2011 4722 y FC(j)15 b FD(\017)2088 4684 y Fx(n)2088 4744 y(i)2160 4722 y FK(=)25 b FD($)s(;)15 b(\017)2411 4736 y FJ(1)2466 4722 y FC(\001)g(\001)g(\001)h FD(\017)2624 4736 y Fx(r)2687 4722 y FK(=)25 b(1)2828 4648 y Ft(\011)2882 4722 y FD(:)125 4906 y FK(There)38 b(is)g(an)h(action)g(of)g(the)g(symmetric)f(group)g FD(S)1962 4920 y Fx(r)2039 4906 y FK(on)2200 4883 y(^)2174 4906 y FD(N)2247 4920 y Fx(n;r)2347 4906 y FK(\()p FD($)s FK(\))h(b)m(y)g(p)s(erm)m(utation)f(of)h(the)g(ele-)125 5027 y(men)m(ts)33 b FD(\017)430 5041 y Fx(i)459 5027 y FK(,)h(and)f(w)m(e)h(denote)h(b)m(y)e FD(N)1334 5041 y Fx(n;r)1434 5027 y FK(\()p FD($)s(;)15 b(\030)t FK(\))32 b(=)1826 5004 y(^)1799 5027 y FD(N)1872 5041 y Fx(n;r)1973 5027 y FK(\()p FD($)s FK(\))p FD(=S)2222 5041 y Fx(r)2294 5027 y FK(the)i(corresp)s(onding)e(unordered)g(set.)125 5135 y(Therefore,)e(the)g(p)s(ossible)e(con\014gurations)i(of)h(eigen)m (v)-5 b(alues)30 b(\()2256 5135 y SDict begin H.S end 2256 5135 a 0 TeXcolorgray FK(8)p 0 TeXcolorgray 2302 5076 a SDict begin H.R end 2302 5076 a 2302 5135 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.8) cvn H.B /ANN pdfmark end 2302 5135 a FK(\))g(are)h(pairs)883 5320 y FD(\024)25 b FK(=)g(\()p FD(\024)1143 5334 y FJ(1)1183 5320 y FD(;)15 b(\024)1275 5334 y FJ(2)1316 5320 y FK(\))25 b FC(2)g FD(M)1550 5334 y Fx(n;m;r)1758 5320 y FK(=)1896 5233 y Ft(G)1854 5424 y Fx($)r Fs(2)p Fx(\026)2004 5432 y Fq(r)2054 5320 y FD(N)2127 5334 y Fx(n;r)2227 5320 y FK(\()p FD($)s FK(\))c FC(\002)f FD(N)2560 5334 y Fx(m;r)2680 5320 y FK(\()p FD($)s FK(\))p FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.19) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(19)p 0 TeXcolorgray 224 266 a FK(Moreo)m(v)m(er,)37 b(consider)c(a)i(partition)d FD(\031)j FK(=)c FC(f)p FK(1)1752 233 y Fx(e)1785 242 y FG(1)1824 266 y FD(;)15 b FK(2)1909 233 y Fx(e)1942 242 y FG(2)1982 266 y FD(;)g(:)g(:)g(:)h(;)f(r)2227 233 y Fx(e)2260 241 y Fq(r)2299 266 y FC(g)34 b FK(of)g FD(r)j FK(with)32 b FD(e)2815 280 y Fx(i)2875 266 y FC(\025)f FK(0)k(for)e(1)f FC(\024)f FD(i)h FC(\024)f FD(r)s FK(,)125 374 y(and)h FD(r)g FK(=)477 305 y Ft(P)573 400 y Fx(i)616 374 y FD(ie)689 388 y Fx(i)718 374 y FK(.)48 b(Let)34 b(us)e(denote)h FD(S)1427 388 y Fx(\031)1504 374 y FK(=)c(\()p FD(S)1695 388 y FJ(1)1734 374 y FK(\))1769 341 y Fx(e)1802 350 y FG(1)1863 374 y FC(\002)22 b FK(\()p FD(S)2047 388 y FJ(2)2086 374 y FK(\))2121 341 y Fx(e)2154 350 y FG(2)2215 374 y FC(\002)f FD(:)15 b(:)g(:)23 b FC(\002)f FK(\()p FD(S)2619 388 y Fx(r)2657 374 y FK(\))2692 341 y Fx(e)2725 349 y Fq(r)2797 374 y FK(as)33 b(a)g(subgroup)e(of)j FD(S)3549 388 y Fx(r)3586 374 y FK(.)125 495 y(W)-8 b(e)27 b(will)d(denote)j(b)m(y)883 472 y(^)856 495 y FD(N)939 462 y Fx(\031)929 517 y(n;r)1030 495 y FK(\()p FD($)s FK(\))g(the)f(set)h(of)g(p)s(oin)m(ts)e(of)1990 472 y(^)1964 495 y FD(N)2037 509 y Fx(n;r)2137 495 y FK(\()p FD($)s FK(\))i(whose)g(stabilizer)d(under)h(the)i(action)125 625 y(of)h FD(S)282 639 y Fx(r)349 625 y FK(is)f(\(conjugated)j(to\))g FD(S)1135 639 y Fx(\031)1181 625 y FK(.)41 b(In)27 b(other)i(w)m(ords,) 1905 602 y(^)1879 625 y FD(N)1962 592 y Fx(\031)1952 647 y(n;r)2052 625 y FK(\()p FD($)s FK(\))h(is)d(the)i(set)g(of)g (tuples)e(of)3111 602 y(^)3084 625 y FD(N)3157 639 y Fx(n;r)3258 625 y FK(\()p FD($)s FK(\))i(with)125 754 y FD(e)167 768 y Fx(i)228 754 y FK(collections)j(of)g FD(i)h FK(equal)f(elemen)m(ts,)i(for)f(1)c FC(\024)g FD(i)g FC(\024)g FD(r)s FK(.)47 b(Denote)34 b(also)e(b)m(y)h FD(N)2783 722 y Fx(\031)2773 777 y(n;r)2873 754 y FK(\()p FD($)s FK(\))d(=)3177 732 y(^)3150 754 y FD(N)3233 722 y Fx(\031)3223 777 y(n;r)3324 754 y FK(\()p FD($)s FK(\))p FD(=S)3573 768 y Fx(r)125 862 y FK(the)g(collection)g(of)h(unordered)d (tuples)i(and)f(set)125 1039 y(\(27\))1171 1039 y SDict begin H.S end 1171 1039 a 1171 1039 a SDict begin 13 H.A end 1171 1039 a 1171 1039 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.27) cvn /DEST pdfmark end 1171 1039 a FD(M)1269 1002 y Fx(\031)1310 1011 y FG(1)1345 1002 y Fx(;\031)1406 1011 y FG(2)1259 1062 y Fx(n;m;r)1469 1039 y FK(=)1607 953 y Ft(G)1565 1144 y Fx($)r Fs(2)p Fx(\026)1715 1152 y Fq(r)1765 1039 y FD(N)1848 1002 y Fx(\031)1889 1011 y FG(1)1838 1062 y Fx(n;r)1938 1039 y FK(\()p FD($)s FK(\))21 b FC(\002)f FD(N)2281 1002 y Fx(\031)2322 1011 y FG(2)2271 1062 y Fx(m;r)2391 1039 y FK(\()p FD($)s FK(\))p FD(:)224 1344 y FK(F)-8 b(or)44 b(giv)m(en)f(partitions)f FD(\031)1130 1358 y FJ(1)1169 1344 y FD(;)15 b(\031)1261 1358 y FJ(2)1300 1344 y FK(,)47 b(if)42 b FD(\024;)15 b(\024)1612 1311 y Fs(0)1683 1344 y FC(2)46 b FD(M)1888 1300 y Fx(\031)1929 1309 y FG(1)1963 1300 y Fx(;\031)2024 1309 y FG(2)1878 1356 y Fx(n;m;r)2105 1344 y FK(then)c(w)m(e)i(ha)m(v)m(e)g([)p FC(R)2795 1311 y FJ(red)2795 1366 y Fx(r)n(;\024)2898 1344 y FK(])i(=)g([)p FC(R)3188 1311 y FJ(red)3188 1372 y Fx(r)n(;\024)3279 1353 y Fr(0)3305 1344 y FK(].)79 b(This)125 1462 y(follo)m(ws)34 b(from)h(\()683 1462 y SDict begin H.S end 683 1462 a 0 TeXcolorgray FK(9)p 0 TeXcolorgray 729 1403 a SDict begin H.R end 729 1403 a 729 1462 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.9) cvn H.B /ANN pdfmark end 729 1462 a FK(\))g(thanks)g(to)i(the)e(form)m(ula)g(in)f(Corollary)2235 1462 y SDict begin H.S end 2235 1462 a 0 TeXcolorgray FK(4.6)p 0 TeXcolorgray 2351 1403 a SDict begin H.R end 2351 1403 a 2351 1462 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.6) cvn H.B /ANN pdfmark end 2351 1462 a FK(,)j(whic)m(h)d(sho)m (ws)h(that)h(the)g(motiv)m(es)125 1569 y(only)27 b(dep)s(end)g(on)h(ho) m(w)h(man)m(y)g(eigen)m(v)-5 b(alues)28 b(are)h(rep)s(eated)f(and)g (not)h(their)f(particular)e(v)-5 b(alues.)40 b(The)125 1677 y(form)m(ula)28 b(only)h(giv)m(es)h(the)f(motiv)m(es)h(under)e (the)i(assumptions)e(of)h(Corollary)2799 1677 y SDict begin H.S end 2799 1677 a 0 TeXcolorgray FK(4.7)p 0 TeXcolorgray 2915 1619 a SDict begin H.R end 2915 1619 a 2915 1677 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.7) cvn H.B /ANN pdfmark end 2915 1677 a FK(,)h(whic)m(h)e(is)g(all)h (that)125 1785 y(w)m(e)g(need.)41 b(Ho)m(w)m(ev)m(er,)31 b(w)m(orking)e(with)f(the)i(diagram)f(\()1988 1785 y SDict begin H.S end 1988 1785 a 0 TeXcolorgray FK(15)p 0 TeXcolorgray 2079 1727 a SDict begin H.R end 2079 1727 a 2079 1785 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.15) cvn H.B /ANN pdfmark end 2079 1785 a FK(\),)h(w)m(e)g(can)g (push-do)m(wn)d(to)j FC(I)7 b FK(\()p FD(\034)j FK(\))30 b(and)f(sho)m(w)125 1893 y(that)i(the)f(classes)g(are)h(equal)f(ev)m (en)h(if)f(w)m(e)g(do)h(not)f(ha)m(v)m(e)i(explicit)d(form)m(ulas.)224 2037 y(The)42 b(ab)s(o)m(v)m(e)i(means)e(that)h(it)f(is)f(enough)h(to)h (compute)g([)p FC(R)2329 2004 y FJ(red)2329 2059 y Fx(r)n(;\024)2431 2037 y FK(])g(for)f(one)h FD(\024)i FC(2)g FD(M)3130 1992 y Fx(\031)3171 2001 y FG(1)3205 1992 y Fx(;\031)3266 2001 y FG(2)3120 2048 y Fx(n;m;r)3346 2037 y FK(\(whic)m(h)125 2148 y(is)38 b(a)j(motiv)m(e)f(indep)s(enden)m(t)e(of)i FD(n)f FK(and)g FD(m)p FK(\))h(and)f(then)h(m)m(ultiply)d(the)j(result) f(b)m(y)h(the)g(n)m(um)m(b)s(er)e(of)125 2256 y(comp)s(onen)m(ts)29 b FC(j)p FD(M)747 2212 y Fx(\031)788 2221 y FG(1)822 2212 y Fx(;\031)883 2221 y FG(2)737 2268 y Fx(n;m;r)921 2256 y FC(j)g FK(\(a)h(com)m(binatorial)f(n)m(um)m(b)s(er)e(dep)s (enden)m(t)i(only)f(on)h FD(n;)15 b(m;)g(r)s FK(\).)40 b(That)29 b(is,)g(\()3485 2256 y SDict begin H.S end 3485 2256 a 0 TeXcolorgray FK(10)p 0 TeXcolorgray 3576 2198 a SDict begin H.R end 3576 2198 a 3576 2256 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.10) cvn H.B /ANN pdfmark end 3576 2256 a FK(\))125 2364 y(and)g(\()336 2364 y SDict begin H.S end 336 2364 a 0 TeXcolorgray FK(7)p 0 TeXcolorgray 382 2306 a SDict begin H.R end 382 2306 a 382 2364 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.7) cvn H.B /ANN pdfmark end 382 2364 a FK(\))i(read)f(as)1238 2508 y([)p FC(R)1340 2471 y FJ(irr)1340 2531 y Fx(r)1419 2508 y FK(])c(=)1585 2422 y Ft(X)1566 2613 y Fx(\031)1607 2622 y FG(1)1641 2613 y Fx(;\031)1702 2622 y FG(2)1751 2508 y FD(c)p FK(\()p FD(\031)1877 2522 y FJ(1)1917 2508 y FD(;)15 b(\031)2009 2522 y FJ(2)2048 2508 y FK(\)[)p FD(R)2178 2471 y FJ(irr)2177 2535 y Fx(\024)p FJ(\()p Fx(\031)2286 2544 y FG(1)2321 2535 y Fx(;\031)2382 2544 y FG(2)2416 2535 y FJ(\))2447 2508 y FK(])p FD(;)125 2753 y FK(where)29 b FD(c)p FK(\()p FD(\031)513 2767 y FJ(1)553 2753 y FD(;)15 b(\031)645 2767 y FJ(2)685 2753 y FK(\))26 b(=)f FC(j)p FD(M)965 2708 y Fx(\031)1006 2717 y FG(1)1040 2708 y Fx(;\031)1101 2717 y FG(2)955 2764 y Fx(n;m;r)1139 2753 y FC(j)p FK(,)31 b(and)f FD(\024)p FK(\()p FD(\031)1536 2767 y FJ(1)1575 2753 y FD(;)15 b(\031)1667 2767 y FJ(2)1707 2753 y FK(\))31 b(is)e(one)i(elemen)m(t)g (in)e FD(M)2571 2708 y Fx(\031)2612 2717 y FG(1)2646 2708 y Fx(;\031)2707 2717 y FG(2)2561 2764 y Fx(n;m;r)2745 2753 y FK(.)224 2896 y(No)m(w)h(w)m(e)g(compute)f(the)g(v)-5 b(alue)29 b(of)g FD(c)p FK(\()p FD(\031)1547 2910 y FJ(1)1587 2896 y FD(;)15 b(\031)1679 2910 y FJ(2)1719 2896 y FK(\))29 b(for)g(some)h(classes)f(of)g(partitions,)f(whic)m(h)g(will)f(co)m(v)m (er)125 3004 y(at)h(least)g(all)f(situation)g(for)g FD(r)h FK(=)d(4.)40 b(W)-8 b(e)29 b(tak)m(e)h FD(k)e FK(=)d Fz(C)51 b FK(in)27 b(the)h(computations)g(b)s(elo)m(w,)f(but)g(the)h (result)125 3112 y(holds)36 b(as)h(w)m(ell)g(for)g(an)g(algebraically)f (closed)i(\014eld)e(of)h(zero)i(c)m(haracteristic,)h(as)e(it)f(only)f (in)m(v)m(olv)m(es)125 3220 y(ro)s(ots)30 b(of)h(unit)m(y)-8 b(.)125 3327 y SDict begin H.S end 125 3327 a 125 3327 a SDict begin 13 H.A end 125 3327 a 125 3327 a SDict begin [/View [/XYZ H.V]/Dest (subsection.6.1) cvn /DEST pdfmark end 125 3327 a 126 x FK(6.1.)46 b FL(P)m(artition)36 b FD(\031)30 b FK(=)c FC(f)p FK(1)1028 3420 y Fx(r)1067 3453 y FC(g)p FL(.)46 b FK(W)-8 b(e)32 b(\014x)e FD(\013)1531 3467 y FJ(1)1571 3453 y FD(;)15 b(:)g(:)g(:)i(;)e(\013)1831 3467 y Fx(n)1909 3453 y FK(the)31 b FD(n)p FK(-th)g(ro)s(ots)g(of)g (unit)m(y)-8 b(,)31 b FD(\013)2917 3467 y Fx(j)2980 3453 y FK(=)c FD(e)3120 3420 y FJ(2)p Fx(\031)r(ij)t(=n)3337 3453 y FK(.)42 b(Since)125 3571 y FD($)37 b FC(2)d FD(\026)387 3585 y Fx(r)424 3571 y FK(,)k(w)m(e)e(can)g(write)f FD($)j FK(=)c FD( )1316 3538 y Fx(a)1394 3571 y FK(for)h(some)h FD(a)f FK(=)f(0)p FD(;)15 b(:)g(:)g(:)i(;)e(r)27 b FC(\000)c FK(1,)38 b(where)e FD( )i FK(=)c FD(e)2989 3538 y FJ(2)p Fx(\031)r(i=r)3164 3571 y FK(.)58 b(Recalling)125 3680 y(\()160 3680 y SDict begin H.S end 160 3680 a 0 TeXcolorgray FK(26)p 0 TeXcolorgray 251 3621 a SDict begin H.R end 251 3621 a 251 3680 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.26) cvn H.B /ANN pdfmark end 251 3680 a FK(\),)31 b(w)m(e)g(tak)m(e)h FD(\020)f FK(=)25 b FD(e)882 3647 y FJ(2)p Fx(\031)r(i=r)r(n)1131 3680 y FK(so)31 b(that)903 3843 y FD(\017)940 3857 y Fx(j)1002 3843 y FK(=)25 b FD(\020)1145 3805 y Fx(a)1186 3843 y FD(\013)1244 3857 y Fx(i)1268 3867 y Fq(j)1305 3843 y FD(;)106 b FK(1)26 b FC(\024)f FD(i)1634 3857 y FJ(1)1674 3843 y FD(;)15 b(:)g(:)g(:)h(;)f(i)1906 3857 y Fx(r)1970 3843 y FC(\024)25 b FD(n;)15 b(i)2192 3857 y Fx(j)2254 3843 y FC(6)p FK(=)25 b FD(i)2381 3858 y Fx(k)2454 3843 y FK(for)30 b FD(j)h FC(6)p FK(=)25 b FD(k)s(:)125 4008 y FK(W)-8 b(e)31 b(in)m(tro)s(duce)e (the)i(m)m(ulti-index)d FD(I)k FK(=)25 b FC(f)p FD(i)1571 4022 y FJ(1)1611 4008 y FD(;)15 b(:)g(:)g(:)i(;)e(i)1844 4022 y Fx(r)1883 4008 y FC(g)30 b FK(and)g(consider)f(the)i(sum)1104 4132 y Fx(n)1067 4160 y Ft(Y)1064 4355 y Fx(j)t FJ(=1)1186 4246 y FK(\(1)21 b(+)f FD(\020)1425 4208 y Fx(a)1466 4246 y FD(\013)1524 4260 y Fx(j)1561 4246 y FD(t)p FK(\))26 b(=)1751 4160 y Ft(X)1754 4355 y Fx(r)r Fs(\025)p FJ(0)1913 4160 y Ft(X)1897 4361 y Fs(j)p Fx(I)5 b Fs(j)p FJ(=)p Fx(r)2076 4246 y FD(\020)2123 4208 y Fx(ar)2198 4246 y FD(\013)2256 4260 y Fx(i)2280 4269 y FG(1)2334 4246 y FC(\001)15 b(\001)g(\001)h FD(\013)2513 4260 y Fx(i)2537 4268 y Fq(r)2576 4246 y FD(t)2609 4208 y Fx(r)2647 4246 y FD(:)125 4513 y FK(W)-8 b(e)34 b(w)m(an)m(t)h(to)f(coun)m(t)h(those)f (pro)s(ducts)e FD(\020)1546 4480 y Fx(ar)1621 4513 y FD(\013)1679 4527 y Fx(i)1703 4536 y FG(1)1757 4513 y FC(\001)15 b(\001)g(\001)h FD(\013)1936 4527 y Fx(i)1960 4535 y Fq(r)2030 4513 y FK(=)30 b(1.)51 b(F)-8 b(or)35 b(this)d(w)m(e)i(mak)m(e)h(use)e(of)h(the)g(\\pro-)125 4621 y(jection")c(op)s(erator)1321 4717 y Fx(n)p Fs(\000)p FJ(1)1322 4744 y Ft(X)1323 4942 y Fx(k)r FJ(=0)1469 4830 y FD(\013)1527 4793 y Fx(k)1595 4830 y FK(=)1691 4702 y Ft(\032)1801 4775 y FD(n;)83 b FK(if)29 b FD(\013)d FK(=)f(1)p FD(;)1801 4883 y FK(0)p FD(;)93 b FK(otherwise)o FD(;)125 5059 y FK(for)30 b FD(\013)25 b FC(2)g FD(\026)488 5073 y Fx(n)535 5059 y FK(.)41 b(Hence)428 5192 y Fx(n)p Fs(\000)p FJ(1)429 5219 y Ft(X)431 5417 y Fx(k)r FJ(=0)616 5192 y Fx(n)580 5219 y Ft(Y)577 5415 y Fx(j)t FJ(=1)699 5306 y FK(\(1)21 b(+)f FD(\020)938 5268 y Fx(ak)1018 5306 y FD(\013)1076 5268 y Fx(k)1076 5328 y(j)1119 5306 y FD(t)p FK(\))25 b(=)1308 5219 y Ft(X)1312 5415 y Fx(r)r Fs(\025)p FJ(0)1471 5219 y Ft(X)1455 5421 y Fs(j)p Fx(I)5 b Fs(j)p FJ(=)p Fx(r)1633 5192 y(n)p Fs(\000)p FJ(1)1634 5219 y Ft(X)1636 5417 y Fx(k)r FJ(=0)1782 5306 y FD(\020)1829 5268 y Fx(ak)r(r)1942 5306 y FK(\()p FD(\013)2035 5320 y Fx(i)2059 5329 y FG(1)2114 5306 y FC(\001)15 b(\001)g(\001)h FD(\013)2293 5320 y Fx(i)2317 5328 y Fq(r)2356 5306 y FK(\))2391 5268 y Fx(k)2434 5306 y FD(t)2467 5268 y Fx(r)2530 5306 y FK(=)2626 5219 y Ft(X)2629 5415 y Fx(r)r Fs(\025)p FJ(0)2772 5306 y FD(n)f FC(j)p FD(N)2950 5268 y Fx(\031)2940 5328 y(n;r)3041 5306 y FK(\()p FD(!)s FK(\))p FC(j)g FD(t)3244 5268 y Fx(r)3282 5306 y FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.20) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(20)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 125 266 a FK(No)m(w)31 b(w)m(e)f(redo)h(the)f(sum)g(as)922 424 y Ft(X)926 620 y Fx(r)r Fs(\025)p FJ(0)1069 511 y FD(n)15 b FC(j)p FD(N)1247 473 y Fx(\034)1237 533 y(n;r)1337 511 y FK(\()p FD(!)s FK(\))p FC(j)p FD(t)1525 473 y Fx(r)1589 511 y FK(=)1685 424 y Ft(X)1701 626 y Fx(d)p Fs(j)p Fx(n)1846 424 y Ft(X)1832 622 y Fx(s)p Fs(2)p Fl(Z)1962 603 y Fr(\003)1962 638 y Fq(e)2046 397 y Fx(n)2010 424 y Ft(Y)2007 620 y Fx(j)t FJ(=1)2144 410 y Ft(\020)2199 511 y FK(1)20 b(+)g(\()p FD(\020)2437 473 y Fx(ad)2515 511 y FD(\013)2573 473 y Fx(d)2573 533 y(j)2614 511 y FK(\))2649 473 y Fx(s)2686 511 y FD(t)2719 410 y Ft(\021)2788 511 y FD(;)125 814 y FK(where)26 b FD(d)p FC(j)p FD(n)p FK(,)i FD(e)d FK(=)g FD(n=d)i FK(and)f Fz(Z)1139 781 y Fs(\003)1139 836 y Fx(e)1201 814 y FK(are)h(the)g(units)f(of)h Fz(Z)1892 828 y Fx(e)1924 814 y FK(,)h(in)m(terpreting)d(that)j Fz(Z)2730 781 y Fs(\003)2730 838 y FJ(1)2790 814 y FK(=)d FC(f)p FK(0)p FC(g)p FK(.)41 b(The)26 b(elemen)m(ts)125 925 y FD(\013)183 892 y Fx(d)183 951 y(j)253 925 y FK(are)j(the)h FD(e)p FK(-th)f(ro)s(ots)h(of)f(unit)m(y)-8 b(,)29 b(rep)s(eated)g FD(d)h FK(times.)40 b(W)-8 b(e)30 b(sub)s(divide)25 b(them)30 b(in)m(to)f(groups.)39 b(Let)30 b(us)125 1051 y(consider)f(the)h (\014rst)g(group)g FD(\013)1139 1018 y Fx(d)1139 1075 y FJ(1)1180 1051 y FD(;)15 b(:)g(:)g(:)h(;)f(\013)1439 1018 y Fx(d)1439 1074 y(e)1481 1051 y FK(.)40 b(The)30 b(p)s(olynomial)125 1321 y(\(28\))1030 1321 y SDict begin H.S end 1030 1321 a 1030 1321 a SDict begin 13 H.A end 1030 1321 a 1030 1321 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.28) cvn /DEST pdfmark end 1030 1321 a 1090 1207 a Fx(e)1048 1234 y Ft(Y)1045 1430 y Fx(j)t FJ(=1)1183 1220 y Ft(\020)1237 1321 y FK(1)21 b(+)f(\()p FD(\020)1476 1283 y Fx(ad)1554 1321 y FD(\013)1612 1283 y Fx(d)1612 1343 y(j)1653 1321 y FK(\))1688 1283 y Fx(s)1725 1321 y FD(t)1758 1220 y Ft(\021)1837 1321 y FK(=)25 b(\(1)c(+)f(\()p FC(\000)p FK(1\))2311 1283 y Fx(e)p FJ(+1)2439 1321 y FD( )2501 1283 y Fx(as)2576 1321 y FD(t)2609 1283 y Fx(e)2645 1321 y FK(\))p FD(;)125 1625 y FK(since)40 b(the)i FD(e)f FK(complex)g(n)m(um)m(b)s(ers)f FC(\000)p FK(\()p FD(\020)1507 1592 y Fx(ad)1584 1625 y FD(\013)1642 1592 y Fx(d)1642 1649 y FJ(1)1683 1625 y FK(\))1718 1592 y Fx(s)1755 1625 y FD(;)15 b(:)g(:)g(:)i(;)e FC(\000)p FK(\()p FD(\020)2110 1592 y Fx(ad)2188 1625 y FD(\013)2246 1592 y Fx(d)2246 1647 y(n)2293 1625 y FK(\))2328 1592 y Fx(s)2406 1625 y FK(are)42 b(exactly)g(those)f FD(z)46 b FK(suc)m(h)41 b(that)125 1736 y FD(z)171 1703 y Fx(e)233 1736 y FK(=)25 b(\()p FC(\000)p FK(1\))515 1703 y Fx(e)553 1736 y FD(\020)600 1703 y Fx(adse)768 1736 y FK(=)g(\()p FC(\000)p FK(1\))1050 1703 y Fx(e)1087 1736 y FD(\020)1134 1703 y Fx(asn)1276 1736 y FK(=)g(\()p FC(\000)p FK(1\))1558 1703 y Fx(e)1596 1736 y FD( )1658 1703 y Fx(as)1733 1736 y FK(.)40 b(The)27 b(other)i(groups)e(giv)m(e)i(the)f(same)h(answ)m(er,)g(so)f(w)m(e)125 1844 y(ha)m(v)m(e)j(to)g(tak)m(e)h(the)f FD(d)p FK(-th)g(p)s(o)m(w)m (er)f(of)g(\()1396 1844 y SDict begin H.S end 1396 1844 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 1488 1786 a SDict begin H.R end 1488 1786 a 1488 1844 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.28) cvn H.B /ANN pdfmark end 1488 1844 a FK(\).)41 b(Putting)29 b(all)h(together,)881 1970 y Ft(X)884 2166 y Fx(r)r Fs(\025)p FJ(0)1027 2056 y FD(n)15 b FC(j)p FD(N)1205 2019 y Fx(\031)1195 2079 y(n;r)1295 2056 y FK(\()p FD($)s FK(\))p FC(j)p FD(t)1501 2019 y Fx(r)1565 2056 y FK(=)1661 1970 y Ft(X)1679 2171 y Fx(e)p Fs(j)p Fx(n)1822 1970 y Ft(X)1808 2167 y Fx(s)p Fs(2)p Fl(Z)1938 2148 y Fr(\003)1938 2184 y Fq(e)1968 2056 y FK(\(1)21 b(+)e(\()p FC(\000)p FK(1\))2345 2019 y Fx(e)p FJ(+1)2473 2056 y FD( )2535 2019 y Fx(as)2610 2056 y FD(t)2643 2019 y Fx(e)2680 2056 y FK(\))2715 2019 y Fx(n=e)2830 2056 y FD(;)125 2354 y FK(and)29 b(taking)i(the)f(term)h FD(t)989 2321 y Fx(r)1026 2354 y FK(,)125 2597 y(\(29\))920 2597 y SDict begin H.S end 920 2597 a 920 2597 a SDict begin 13 H.A end 920 2597 a 920 2597 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.29) cvn /DEST pdfmark end 920 2597 a FC(j)p FD(N)1028 2560 y Fx(\031)1018 2620 y(n;r)1119 2597 y FK(\()p FD($)s FK(\))p FC(j)26 b FK(=)1428 2536 y(1)p 1424 2576 55 4 v 1424 2660 a FD(n)1538 2511 y Ft(X)1503 2712 y Fx(e)p Fs(j)p Fx(n;e)p Fs(j)p Fx(r)1734 2511 y Ft(X)1719 2708 y Fx(s)p Fs(2)p Fl(Z)1849 2689 y Fr(\003)1849 2725 y Fq(e)1879 2597 y FK(\()p FC(\000)p FK(1\))2065 2560 y Fx(r)r FJ(+)p Fx(r)r(=e)2261 2469 y Ft(\022)2328 2536 y FD(n=e)2333 2660 y(r)s(=e)2470 2469 y Ft(\023)2537 2597 y FD( )2599 2560 y Fx(asr)r(=e)2791 2597 y FD(:)125 2915 y FK(Here)35 b FD( )406 2882 y Fx(r)r(=e)548 2915 y FK(is)f(a)h(primitiv)m(e)e FD(e)p FK(-ro)s(ot)j(of)g(unit)m(y)-8 b(,)36 b(and)e FD( )2003 2882 y Fx(sr)r(=e)2177 2915 y FK(runs)g(o)m(v)m(er)i(all)e(primitiv)m(e)f FD(e)p FK(-th)j(ro)s(ots)f(of)125 3022 y(unit)m(y)29 b FD(\026)417 2989 y Fs(\003)417 3045 y Fx(e)456 3022 y FK(.)41 b(The)30 b(\014nal)f(form)m(ula)g(is:)125 3283 y(\(30\))1011 3283 y SDict begin H.S end 1011 3283 a 1011 3283 a SDict begin 13 H.A end 1011 3283 a 1011 3283 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.30) cvn /DEST pdfmark end 1011 3283 a FC(j)p FD(N)1119 3246 y Fx(\031)1109 3306 y(n;r)1209 3283 y FK(\()p FD($)s FK(\))p FC(j)d FK(=)1539 3197 y Ft(X)1504 3398 y Fx(e)p Fs(j)p Fx(n;e)p Fs(j)p Fx(r)1730 3222 y FK(\()p FC(\000)p FK(1\))1916 3189 y Fx(r)r FJ(+)p Fx(r)r(=e)p 1730 3262 382 4 v 1894 3345 a FD(n)2122 3155 y Ft(\022)2189 3222 y FD(n=e)2194 3345 y(r)s(=e)2331 3155 y Ft(\023)2430 3197 y(X)2413 3390 y Fx(\027)t Fs(2)p Fx(\026)2541 3371 y Fr(\003)2541 3407 y Fq(e)2593 3283 y FD(\027)2644 3246 y Fx(a)2700 3283 y FD(:)125 3576 y SDict begin H.S end 125 3576 a 125 3576 a SDict begin 13 H.A end 125 3576 a 125 3576 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.1) cvn /DEST pdfmark end 125 3576 a 0 TeXcolorgray FL(Remark)34 b(6.1.)p 0 TeXcolorgray 42 w FK(The)c(last)g(term)h(is)e(can)i(b)s(e)e(computed)i(via)f(the)g (M\177)-45 b(obius)29 b(in)m(v)m(ersion)g(form)m(ula:)1314 3697 y Ft(X)1298 3891 y Fx(\027)t Fs(2)p Fx(\026)1426 3872 y Fr(\003)1426 3907 y Fq(e)1477 3783 y FD(\027)1528 3746 y Fx(a)1595 3783 y FK(=)1691 3697 y Ft(X)1712 3898 y Fx(y)r Fs(j)p Fx(e)1837 3783 y FD(\026)p FK(\()p FD(y)s FK(\))15 b(SR\()p FD(e=y)s FK(;)g FD(a)p FK(\))125 4086 y(where)31 b FD(\026)p FK(\()p FD(y)s FK(\))h(is)e(the)i(M\177)-45 b(obius)31 b(function,)g(and)g(SR)o(\()p FD(e)p FK(;)15 b FD(a)p FK(\))29 b(=)2172 4018 y Ft(P)2268 4113 y Fx(\027)t Fs(2)p Fx(\026)2396 4121 y Fq(e)2449 4086 y FD(\027)2500 4053 y Fx(a)2542 4086 y FK(.)44 b(So)32 b(SR)o(\()p FD(e)p FK(;)15 b FD(a)p FK(\))29 b(=)e FD(e)32 b FK(when)f FD(e)p FC(j)p FD(a)125 4197 y FK(and)e(0)i(otherwise.)125 4279 y SDict begin H.S end 125 4279 a 125 4279 a SDict begin 13 H.A end 125 4279 a 125 4279 a SDict begin [/View [/XYZ H.V]/Dest (subsection.6.2) cvn /DEST pdfmark end 125 4279 a 143 x FK(6.2.)46 b FL(P)m(artition)35 b FD(\031)813 4389 y Fs(0)862 4422 y FK(=)958 4348 y Ft(\010)1011 4422 y FK(1)1056 4389 y Fx(r)r Fs(\000)p FJ(2)1185 4422 y FD(;)15 b FK(2)1270 4389 y FJ(1)1310 4348 y Ft(\011)1363 4422 y FL(.)45 b FK(W)-8 b(e)32 b(start)f(with)e(the)i(the)f(sum)784 4585 y Fx(n)739 4613 y Ft(X)742 4807 y Fx(p)p FJ(=1)886 4699 y FD(\020)933 4661 y FJ(2)p Fx(a)1009 4699 y FD(\013)1067 4661 y FJ(2)1067 4721 y Fx(p)1107 4699 y FD(t)1140 4661 y FJ(2)1198 4613 y Ft(Y)1195 4810 y Fx(j)t Fs(6)p FJ(=)p Fx(p)1317 4699 y FK(\(1)21 b(+)f FD(\020)1556 4661 y Fx(a)1597 4699 y FD(\013)1655 4713 y Fx(j)1692 4699 y FD(t)p FK(\))25 b(=)1881 4613 y Ft(X)1885 4808 y Fx(r)r Fs(\025)p FJ(0)2084 4613 y Ft(X)2038 4801 y Fr(j)p Fq(I)t Fr(j)p FG(=)p Fq(r)r Fr(\000)p FG(2)2098 4860 y Fq(p)7 b(=)-39 b Fr(2)o Fq(I)2287 4699 y FD(\020)2334 4661 y Fx(ar)2409 4699 y FD(\013)2467 4661 y FJ(2)2467 4721 y Fx(p)2507 4699 y FD(\013)2565 4713 y Fx(i)2589 4722 y FG(3)2643 4699 y FD(:)15 b(:)g(:)i(\013)2823 4713 y Fx(i)2847 4721 y Fq(r)2885 4699 y FD(t)2918 4661 y Fx(r)2971 4699 y FD(:)125 5036 y FK(T)-8 b(aking)30 b(the)g(pro)5 b(jection)30 b(op)s(erator)h(as)f(b)s(efore,)812 5199 y Fx(n)p Fs(\000)p FJ(1)813 5227 y Ft(X)814 5424 y Fx(k)r FJ(=0)1004 5199 y Fx(n)960 5227 y Ft(X)963 5421 y Fx(p)p FJ(=1)1107 5313 y FD(\020)1154 5275 y FJ(2)p Fx(ak)1268 5313 y FD(\013)1326 5275 y FJ(2)p Fx(k)1326 5335 y(p)1405 5313 y FD(t)1438 5275 y FJ(2)1496 5227 y Ft(Y)1492 5424 y Fx(j)t Fs(6)p FJ(=)p Fx(p)1615 5313 y FK(\(1)21 b(+)f FD(\020)1854 5275 y Fx(ak)1934 5313 y FD(\013)1992 5275 y Fx(k)1992 5335 y(j)2035 5313 y FD(t)p FK(\))25 b(=)2224 5227 y Ft(X)2228 5423 y Fx(r)r Fs(\025)p FJ(0)2371 5313 y FD(n)15 b FC(j)p FD(N)2549 5275 y Fx(\031)2592 5252 y Fr(0)2539 5335 y Fx(n;r)2639 5313 y FK(\()p FD($)s FK(\))p FC(j)g FD(t)2860 5275 y Fx(r)2899 5313 y FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 21 21 TeXDict begin HPSdict begin 21 20 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.21) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(21)p 0 TeXcolorgray 125 266 a FK(No)m(w)31 b(w)m(e)f(redo)h(the)f(sum,)g(using)f(that)i(\(1)21 b(+)f FD(\020)1716 233 y Fx(ak)1795 266 y FD(\013)1853 233 y Fx(k)1853 288 y(p)1897 266 y FD(t)p FK(\))1965 233 y Fs(\000)p FJ(1)2084 266 y FK(=)2188 197 y Ft(P)2180 350 y Fx(l)q Fs(\025)p FJ(0)2292 266 y FK(\()p FC(\000)p FK(1\))2478 233 y Fx(l)2505 266 y FD(\020)2552 233 y Fx(l)q(ak)2654 266 y FD(\013)2712 233 y Fx(l)q(k)2712 288 y(p)2777 266 y FD(t)2810 233 y Fx(l)2835 266 y FK(,)125 1117 y(\(31\))472 1117 y SDict begin H.S end 472 1117 a 472 1117 a SDict begin 13 H.A end 472 1117 a 472 1117 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.31) cvn /DEST pdfmark end 472 1117 a -619 x Ft(X)476 694 y Fx(r)r Fs(\025)p FJ(0)619 584 y FD(n)15 b FC(j)p FD(N)797 546 y Fx(\031)840 523 y Fr(0)787 606 y Fx(n;r)887 584 y FK(\()p FD($)s FK(\))p FC(j)g FD(t)1108 546 y Fx(r)1172 584 y FK(=)1268 470 y Fx(n)p Fs(\000)p FJ(1)1269 498 y Ft(X)1270 695 y Fx(k)r FJ(=0)1460 470 y Fx(n)1416 498 y Ft(X)1419 692 y Fx(p)p FJ(=1)1593 470 y Fs(1)1563 498 y Ft(X)1572 695 y Fx(l)q FJ(=2)1694 584 y FK(\()p FC(\000)p FK(1\))1880 546 y Fx(l)1907 584 y FD(\020)1954 546 y Fx(l)q(ak)2055 584 y FD(\013)2113 546 y Fx(l)q(k)2113 606 y(p)2178 584 y FD(t)2211 546 y Fx(l)2292 470 y(n)2255 498 y Ft(Y)2252 693 y Fx(j)t FJ(=1)2375 584 y FK(\(1)21 b(+)f FD(\020)2614 546 y Fx(ak)2693 584 y FD(\013)2751 546 y Fx(k)2751 606 y(j)2795 584 y FD(t)p FK(\))912 902 y(=)1053 788 y Fx(n)1008 815 y Ft(X)1011 1010 y Fx(p)p FJ(=1)1185 788 y Fs(1)1155 815 y Ft(X)1164 1013 y Fx(l)q FJ(=2)1301 815 y Ft(X)1318 1017 y Fx(d)p Fs(j)p Fx(n)1462 815 y Ft(X)1448 1013 y Fx(s)p Fs(2)p Fl(Z)1578 994 y Fr(\003)1578 1029 y Fq(e)1608 902 y FK(\()p FC(\000)p FK(1\))1794 864 y Fx(l)1821 902 y FD(\020)1868 864 y Fx(l)q(ads)1999 902 y FD(\013)2057 864 y Fx(l)q(ds)2057 924 y(p)2153 902 y FD(t)2186 864 y Fx(l)2267 788 y(n)2230 815 y Ft(Y)2227 1011 y Fx(j)t FJ(=1)2349 902 y FK(\(1)h(+)f(\()p FD(\020)2623 864 y Fx(ad)2701 902 y FD(\013)2759 864 y Fx(d)2759 924 y(j)2800 902 y FK(\))2835 864 y Fx(s)2872 902 y FD(t)p FK(\))912 1264 y(=)1039 1150 y Fs(1)1008 1178 y Ft(X)1018 1375 y Fx(l)q FJ(=2)1155 1178 y Ft(X)1173 1379 y Fx(e)p Fs(j)p Fx(n)1316 1178 y Ft(X)1301 1375 y Fx(s)p Fs(2)p Fl(Z)1431 1356 y Fr(\003)1431 1391 y Fq(e)1461 1264 y FK(\()p FC(\000)p FK(1\))1647 1226 y Fx(l)1674 1264 y FD(\020)1721 1226 y Fx(l)q(ads)1868 1081 y Ft(0)1868 1245 y(@)1992 1150 y Fx(n)1948 1178 y Ft(X)1950 1372 y Fx(p)p FJ(=1)2094 1264 y FD(\013)2152 1226 y Fx(l)q(ds)2152 1286 y(p)2248 1081 y Ft(1)2248 1245 y(A)2342 1264 y FD(t)2375 1226 y Fx(l)2401 1264 y FK(\(1)h(+)f(\()p FC(\000)p FK(1\))2779 1226 y Fx(e)p FJ(+1)2907 1264 y FD( )2969 1226 y Fx(as)3044 1264 y FD(t)3077 1226 y Fx(e)3113 1264 y FK(\))3148 1226 y Fx(n=e)912 1591 y FK(=)25 b FD(n)1078 1505 y Ft(X)1096 1706 y Fx(e)p Fs(j)p Fx(n)1239 1505 y Ft(X)1224 1702 y Fx(s)p Fs(2)p Fl(Z)1354 1683 y Fr(\003)1354 1719 y Fq(e)1443 1478 y Fs(1)1413 1505 y Ft(X)1430 1692 y Fq(l)p Fr(\025)p FG(2)1410 1748 y Fq(n)p Fr(j)p Fq(lds)1558 1591 y FK(\()p FC(\000)p FK(1\))1744 1554 y Fx(l)1771 1591 y FD(\020)1818 1554 y Fx(l)q(ads)1950 1591 y FD(t)1983 1554 y Fx(l)2008 1591 y FK(\(1)d(+)d(\()p FC(\000)p FK(1\))2386 1554 y Fx(e)p FJ(+1)2514 1591 y FD( )2576 1554 y Fx(as)2651 1591 y FD(t)2684 1554 y Fx(e)2721 1591 y FK(\))2756 1554 y Fx(n=e)2871 1591 y FD(:)125 1886 y FK(Here)30 b FD(n)p FC(j)p FD(l)r(ds)g FK(is)g(equiv)-5 b(alen)m(t)30 b(to)h FD(e)p FC(j)p FD(l)r FK(.)41 b(The)30 b(term)g(with)f FD(e)d FK(=)f(1)31 b(is)827 2053 y FD(n)897 1967 y Ft(X)907 2164 y Fx(l)q Fs(\025)p FJ(2)1028 2053 y FK(\()p FC(\000)p FK(1\))1214 2015 y Fx(l)1241 2053 y FD(\020)1288 2015 y Fx(l)q(ans)1427 2053 y FD(t)1460 2015 y Fx(l)1486 2053 y FK(\(1)21 b(+)e FD( )1739 2015 y Fx(as)1814 2053 y FD(t)p FK(\))1882 2015 y Fx(n)1955 2053 y FK(=)25 b FD(n )2168 2015 y FJ(2)p Fx(as)2277 2053 y FD(t)2310 2015 y FJ(2)2349 2053 y FK(\(1)c(+)f FD( )2603 2015 y Fx(as)2678 2053 y FD(t)p FK(\))2746 2015 y Fx(n)p Fs(\000)p FJ(1)2883 2053 y FD(:)125 2296 y FK(The)29 b(term)i(with)e FD(e)c FC(\025)g FK(2)31 b(reduces)f(to)727 2452 y FD(n)797 2366 y Ft(X)815 2567 y Fx(e)p Fs(j)p Fx(n)958 2366 y Ft(X)943 2563 y Fx(s)p Fs(2)p Fl(Z)1073 2544 y Fr(\003)1073 2580 y Fq(e)1118 2366 y Ft(X)1147 2567 y Fx(e)p Fs(j)p Fx(l)1250 2452 y FK(\()p FC(\000)p FK(1\))1436 2415 y Fx(l)1463 2452 y FD(\020)1510 2415 y Fx(l)q(ads)1641 2452 y FD(t)1674 2415 y Fx(l)1700 2452 y FK(\(1)21 b(+)f(\()p FC(\000)p FK(1\))2078 2415 y Fx(e)p FJ(+1)2206 2452 y FD( )2268 2415 y Fx(as)2343 2452 y FD(t)2376 2415 y Fx(e)2413 2452 y FK(\))2448 2415 y Fx(n=e)1128 2724 y FK(=)25 b FD(n)1294 2637 y Ft(X)1312 2839 y Fx(e)p Fs(j)p Fx(n)1455 2637 y Ft(X)1441 2835 y Fx(s)p Fs(2)p Fl(Z)1571 2816 y Fr(\003)1571 2851 y Fq(e)1601 2724 y FK(\()p FC(\000)p FK(1\))1787 2686 y Fx(e)1824 2724 y FD( )1886 2686 y Fx(as)1961 2724 y FD(t)1994 2686 y Fx(e)2031 2724 y FK(\(1)c(+)f(\()p FC(\000)p FK(1\))2409 2686 y Fx(e)p FJ(+1)2537 2724 y FD( )2599 2686 y Fx(as)2673 2724 y FD(t)2706 2686 y Fx(e)2743 2724 y FK(\))2778 2686 y Fx(n=e)p Fs(\000)p FJ(1)2984 2724 y FD(:)125 2981 y FK(T)-8 b(aking)30 b(the)g(term)h FD(t)835 2948 y Fx(r)903 2981 y FK(in)e(\()1044 2981 y SDict begin H.S end 1044 2981 a 0 TeXcolorgray FK(31)p 0 TeXcolorgray 1135 2923 a SDict begin H.R end 1135 2923 a 1135 2981 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.31) cvn H.B /ANN pdfmark end 1135 2981 a FK(\),)i(and)f(recalling)f (that)i FD( )2026 2948 y Fx(r)2089 2981 y FK(=)25 b(1,)31 b(w)m(e)g(ha)m(v)m(e)125 3184 y(\(32\))664 3184 y SDict begin H.S end 664 3184 a 664 3184 a SDict begin 13 H.A end 664 3184 a 664 3184 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.32) cvn /DEST pdfmark end 664 3184 a FC(j)p FD(N)772 3147 y Fx(\031)815 3123 y Fr(0)762 3207 y Fx(n;r)863 3184 y FK(\()p FD($)s FK(\))p FC(j)26 b FK(=)1158 3056 y Ft(\022)1225 3123 y FD(n)20 b FC(\000)g FK(1)1231 3247 y FD(r)i FC(\000)e FK(2)1436 3056 y Ft(\023)1523 3184 y FC(\000)1651 3098 y Ft(X)1624 3287 y Fq(e)p Fr(j)p Fq(n;e)p Fr(j)p Fq(r)1662 3343 y(e)p Fr(\025)p FG(2)1848 3098 y Ft(X)1833 3295 y Fx(s)p Fs(2)p Fl(Z)1963 3276 y Fr(\003)1963 3312 y Fq(e)1993 3184 y FK(\()p FC(\000)p FK(1\))2179 3147 y Fx(r)r FJ(+)p Fx(r)r(=e)2375 3056 y Ft(\022)2442 3123 y FD(n=e)g FC(\000)g FK(1)2447 3247 y FD(r)s(=e)h FC(\000)f FK(1)2741 3056 y Ft(\023)2808 3184 y FD( )2870 3147 y Fx(asr)r(=e)3046 3184 y FD(:)125 3494 y SDict begin H.S end 125 3494 a 125 3494 a SDict begin 13 H.A end 125 3494 a 125 3494 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.2) cvn /DEST pdfmark end 125 3494 a 0 TeXcolorgray FL(Remark)41 b(6.2.)p 0 TeXcolorgray 46 w FK(There)36 b(is)f(an)i(alternativ)m(e)g(w)m(a)m(y)g(of)g(pro)m (ving)f(the)g(expression)g(\()2974 3494 y SDict begin H.S end 2974 3494 a 0 TeXcolorgray FK(32)p 0 TeXcolorgray 3065 3436 a SDict begin H.R end 3065 3436 a 3065 3494 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.32) cvn H.B /ANN pdfmark end 3065 3494 a FK(\).)59 b(W)-8 b(rite)37 b FD(L)f FK(=)125 3609 y FC(j)p FD(N)233 3576 y Fx(\031)223 3632 y(n;r)323 3609 y FK(\()p FD($)s FK(\))p FC(j)31 b FK(and)f FD(L)766 3576 y Fs(0)814 3609 y FK(=)25 b FC(j)p FD(N)1018 3576 y Fx(\031)1061 3553 y Fr(0)1008 3632 y Fx(n;r)1109 3609 y FK(\()p FD($)s FK(\))p FC(j)p FK(.)42 b(W)-8 b(e)31 b(start)g(with)e(the)i(expression)1291 3731 y Fx(n)1246 3758 y Ft(X)1249 3952 y Fx(p)p FJ(=1)1378 3844 y FK(\(1)21 b FC(\000)f FD(\020)1617 3807 y FJ(2)p Fx(a)1693 3844 y FD(\013)1751 3807 y FJ(2)1751 3867 y Fx(p)1791 3844 y FD(t)1824 3807 y FJ(2)1863 3844 y FK(\))1913 3758 y Ft(Y)1915 3956 y Fx(i)p Fs(6)p FJ(=)p Fx(p)2030 3844 y FK(\(1)h(+)f FD(\020)2269 3807 y Fx(a)2310 3844 y FD(\013)2368 3858 y Fx(i)2396 3844 y FD(t)p FK(\))p FD(:)125 4102 y FK(The)32 b(terms)h(with)f FD(\013)838 4069 y FJ(2)838 4125 y Fx(p)910 4102 y FK(con)m(tribute)h FC(\000)p FD(L)1483 4069 y Fs(0)1521 4102 y FD(t)1554 4069 y Fx(r)1592 4102 y FK(,)h(but)e(there)h(are)g(terms)2465 4034 y Ft(P)2561 4060 y Fx(n)2561 4129 y(p)p FJ(=1)2706 4034 y Ft(Q)2792 4129 y Fx(i)p Fs(6)p FJ(=)p Fx(p)2911 4102 y FK(\(1)22 b(+)g FD(\020)3153 4069 y Fx(a)3194 4102 y FD(\013)3252 4116 y Fx(i)3281 4102 y FD(t)p FK(\))33 b(whic)m(h)125 4218 y(con)m(tribute)j(with)g(\()p FD(n)25 b FC(\000)f FD(r)s FK(\))p FD(L)15 b(t)1181 4185 y Fx(r)1219 4218 y FK(.)60 b(The)37 b(co)s(e\016cien)m(t)h FD(n)24 b FC(\000)g FD(r)40 b FK(comes)e(from)e(the)i(fact)g(that)f FD(\013)3291 4232 y Fx(i)3315 4241 y FG(1)3369 4218 y FC(\001)15 b(\001)g(\001)h FD(\013)3548 4232 y Fx(i)3572 4240 y Fq(r)125 4326 y FK(app)s(ears)31 b(in)g(the)h FD(n)21 b FC(\000)g FD(r)34 b FK(summands)c(for)i FD(p)c FC(2)g(f)p FK(1)p FD(;)15 b(:)g(:)g(:)i(;)e(n)p FC(g)22 b(\000)f(f)p FD(i)2310 4340 y FJ(1)2350 4326 y FD(;)15 b(:)g(:)g(:)i(;)e(i)2583 4340 y Fx(r)2622 4326 y FC(g)p FK(.)46 b(No)m(w)33 b(w)m(e)f(pro)5 b(ject)33 b(as)f(w)m(e)125 4434 y(ha)m(v)m(e)f(done)f(b)s(efore,)h(so)f(that)h FD(n)p FK(\()p FC(\000)p FD(L)1381 4401 y Fs(0)1424 4434 y FK(+)20 b(\()p FD(n)g FC(\000)g FD(r)s FK(\))p FD(L)p FK(\))30 b(equals)g(the)h FD(t)2388 4401 y Fx(r)2426 4434 y FK(-co)s(e\016cien)m (t)g(of)399 4543 y Fx(n)355 4570 y Ft(X)356 4768 y Fx(k)r FJ(=1)546 4543 y Fx(n)501 4570 y Ft(X)504 4765 y Fx(p)p FJ(=1)633 4657 y FK(\(1)21 b FC(\000)f FD(\020)872 4619 y FJ(2)p Fx(ak)987 4657 y FD(\013)1045 4619 y FJ(2)p Fx(k)1045 4679 y(p)1123 4657 y FD(t)1156 4619 y FJ(2)1195 4657 y FK(\))1245 4570 y Ft(Y)1247 4768 y Fx(i)p Fs(6)p FJ(=)p Fx(p)1362 4657 y FK(\(1)h(+)f FD(\020)1601 4619 y Fx(ak)1680 4657 y FD(\013)1738 4619 y Fx(k)1738 4679 y(i)1782 4657 y FD(t)p FK(\))25 b(=)2015 4543 y Fx(n)1971 4570 y Ft(X)1972 4768 y Fx(k)r FJ(=1)2162 4543 y Fx(n)2117 4570 y Ft(X)2120 4765 y Fx(p)p FJ(=1)2249 4657 y FK(\(1)c FC(\000)f FD(\020)2488 4619 y Fx(ak)2567 4657 y FD(\013)2625 4619 y Fx(k)2625 4679 y(p)2668 4657 y FD(t)p FK(\))2788 4543 y Fx(n)2752 4570 y Ft(Y)2753 4766 y Fx(i)p FJ(=1)2868 4657 y FK(\(1)h(+)f FD(\020)3107 4619 y Fx(ak)3187 4657 y FD(\013)3245 4619 y Fx(k)3245 4679 y(i)3288 4657 y FD(t)p FK(\))p FD(:)125 4921 y FK(The)43 b(\014rst)f(summand)g(giv)m (es)i(the)g(con)m(tribution)e FD(n)1945 4888 y FJ(2)1999 4853 y Ft(P)2095 4948 y Fx(e)p Fs(j)p Fx(n)2209 4853 y Ft(P)2305 4948 y Fx(s)p Fs(2)p Fl(Z)2435 4929 y Fr(\003)2435 4965 y Fq(e)2470 4921 y FK(\(1)29 b(+)g(\()p FC(\000)p FK(1\))2865 4888 y Fx(e)2903 4921 y FD( )2965 4888 y Fx(as)3039 4921 y FD(t)3072 4888 y Fx(e)p FJ(+1)3199 4921 y FK(\))3234 4888 y Fx(n=e)3350 4921 y FK(.)80 b(The)125 5049 y(co)s(e\016cien)m(t)32 b(of)g FD(t)688 5016 y Fx(r)757 5049 y FK(is)e(again)i FD(n)1148 5016 y FJ(2)1187 5049 y FD(L)p FK(.)44 b(The)30 b(second)i(summand)e(only)g(con)m(tributes)h (for)h FD(k)e FK(=)d FD(n)k FK(\(that)h(is,)125 5157 y(when)d FD(e)c FK(=)g(1\),)32 b(and)e(so)g(pro)s(duces)f FC(\000)p FD( )1465 5124 y Fx(a)1507 5157 y FD(nt)p FK(\(1)20 b(+)g FD( )1848 5124 y Fx(a)1891 5157 y FD(t)p FK(\))1959 5124 y Fx(n)2006 5157 y FK(.)40 b(Altogether)32 b(this)d(giv)m(es)i (the)f(equalit)m(y)1039 5360 y FC(\000)p FD(nL)1227 5322 y Fs(0)1270 5360 y FK(+)20 b FD(n)p FK(\()p FD(n)f FC(\000)h FD(r)s FK(\))p FD(L)25 b FK(=)g FD(n)1933 5322 y FJ(2)1972 5360 y FD(L)20 b FC(\000)g FD(n )2262 5322 y Fx(ar)2338 5232 y Ft(\022)2477 5298 y FD(n)2405 5422 y(r)i FC(\000)e FK(1)2605 5232 y Ft(\023)2672 5360 y FD(;)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 22 22 TeXDict begin HPSdict begin 22 21 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.22) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(22)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 125 266 a FK(whic)m(h)29 b(is)g(rewritten,)h(using)f(that)i FD( )1393 233 y Fx(r)1456 266 y FK(=)25 b(1,)31 b(as)125 464 y(\(33\))1442 464 y SDict begin H.S end 1442 464 a 1442 464 a SDict begin 13 H.A end 1442 464 a 1442 464 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.33) cvn /DEST pdfmark end 1442 464 a FD(L)1504 427 y Fs(0)1552 464 y FK(=)25 b FC(\000)p FD(r)s(L)19 b FK(+)1935 336 y Ft(\022)2075 403 y FD(n)2002 527 y(r)k FC(\000)d FK(1)2202 336 y Ft(\023)2269 464 y FD(:)125 668 y FK(Finally)28 b(w)m(e)j(use)f(the)g(expression)f(of)i FD(L)f FK(in)f(\()1656 668 y SDict begin H.S end 1656 668 a 0 TeXcolorgray FK(29)p 0 TeXcolorgray 1748 609 a SDict begin H.R end 1748 609 a 1748 668 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.29) cvn H.B /ANN pdfmark end 1748 668 a FK(\),)1095 884 y FD(L)c FK(=)1313 798 y Ft(X)1278 999 y Fx(e)p Fs(j)p Fx(n;e)p Fs(j)p Fx(r)1504 823 y FK(\()p FC(\000)p FK(1\))1690 790 y Fx(r)r FJ(+)p Fx(r)r(=e)p 1504 863 382 4 v 1668 947 a FD(n)1896 756 y Ft(\022)1963 823 y FD(n=e)1968 947 y(r)s(=e)2105 756 y Ft(\023)2201 798 y(X)2187 995 y Fx(s)p Fs(2)p Fl(Z)2317 976 y Fr(\003)2317 1012 y Fq(e)2362 884 y FD( )2424 847 y Fx(asr)r(=e)2616 884 y FD(:)125 1142 y FK(The)k(co)s(e\016cien)m(t)j (corresp)s(onding)c(to)j FD(e)26 b FK(=)f(1)30 b(in)f(\()1811 1142 y SDict begin H.S end 1811 1142 a 0 TeXcolorgray FK(33)p 0 TeXcolorgray 1903 1084 a SDict begin H.R end 1903 1084 a 1903 1142 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.33) cvn H.B /ANN pdfmark end 1903 1142 a FK(\))h(is)1233 1346 y FC(\000)1319 1285 y FD(r)p 1314 1325 55 4 v 1314 1408 a(n)1378 1218 y Ft(\022)1445 1285 y FD(n)1451 1408 y(r)1500 1218 y Ft(\023)1587 1346 y FK(+)1678 1218 y Ft(\022)1817 1285 y FD(n)1745 1408 y(r)22 b FC(\000)e FK(1)1945 1218 y Ft(\023)2037 1346 y FK(=)2133 1218 y Ft(\022)2200 1285 y FD(n)g FC(\000)f FK(1)2205 1408 y FD(r)k FC(\000)d FK(2)2411 1218 y Ft(\023)2478 1346 y FD(:)125 1544 y FK(Hence)801 1687 y FD(L)863 1649 y Fs(0)911 1687 y FK(=)1007 1559 y Ft(\022)1074 1625 y FD(n)g FC(\000)f FK(1)1079 1749 y FD(r)k FC(\000)d FK(2)1285 1559 y Ft(\023)1372 1687 y FC(\000)1499 1601 y Ft(X)1473 1789 y Fq(e)p Fr(j)p Fq(n;e)p Fr(j)p Fq(r)1511 1846 y(e)p Fr(\025)p FG(2)1667 1687 y FK(\()p FC(\000)p FK(1\))1853 1649 y Fx(r)r FJ(+)p Fx(r)r(=e)2049 1559 y Ft(\022)2116 1625 y FD(n=e)g FC(\000)g FK(1)2121 1749 y FD(r)s(=e)h FC(\000)f FK(1)2415 1559 y Ft(\023)2511 1601 y(X)2497 1798 y Fx(s)p Fs(2)p Fl(Z)2627 1779 y Fr(\003)2627 1814 y Fq(e)2672 1687 y FD( )2734 1649 y Fx(asr)r(=e)2910 1687 y FD(;)125 1988 y FK(using)28 b(that)559 1915 y Ft(\000)601 1945 y Fx(n=e)606 2022 y(r)r(=e)712 1915 y Ft(\001)779 1988 y FK(=)885 1952 y Fx(n)p 885 1967 43 4 v 890 2020 a(r)938 1915 y Ft(\000)979 1945 y Fx(n=e)p Fs(\000)p FJ(1)984 2022 y Fx(r)r(=e)p Fs(\000)p FJ(1)1180 1915 y Ft(\001)1222 1988 y FK(.)125 2095 y SDict begin H.S end 125 2095 a 125 2095 a SDict begin 13 H.A end 125 2095 a 125 2095 a SDict begin [/View [/XYZ H.V]/Dest (subsection.6.3) cvn /DEST pdfmark end 125 2095 a 128 x FK(6.3.)46 b FL(P)m(artition)g FD(\031)824 2190 y Fs(00)908 2223 y FK(=)1020 2149 y Ft(\010)1073 2223 y FK(1)1118 2190 y Fx(r)r Fs(\000)p FJ(3)1246 2223 y FD(;)15 b FK(3)1331 2190 y FJ(1)1372 2149 y Ft(\011)1425 2223 y FL(.)45 b FK(W)-8 b(orking)40 b(as)g(in)f(Remark)2463 2223 y SDict begin H.S end 2463 2223 a 0 TeXcolorgray FK(6.2)p 0 TeXcolorgray 2580 2164 a SDict begin H.R end 2580 2164 a 2580 2223 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6.2) cvn H.B /ANN pdfmark end 2580 2223 a 39 w FK(for)h FD(L)2830 2190 y Fs(00)2913 2223 y FK(=)h FC(j)p FD(N)3133 2190 y Fx(\031)3176 2166 y Fr(00)3123 2245 y Fx(n;r)3224 2223 y FK(\()p FD($)s FK(\))p FC(j)p FK(,)i(and)125 2331 y(starting)30 b(with)1291 2397 y Fx(n)1246 2425 y Ft(X)1249 2619 y Fx(p)p FJ(=1)1378 2511 y FK(\(1)21 b(+)f FD(\020)1617 2473 y FJ(3)p Fx(a)1693 2511 y FD(\013)1751 2473 y FJ(3)1751 2533 y Fx(p)1791 2511 y FD(t)1824 2473 y FJ(3)1863 2511 y FK(\))1913 2425 y Ft(Y)1915 2622 y Fx(i)p Fs(6)p FJ(=)p Fx(p)2030 2511 y FK(\(1)h(+)f FD(\020)2269 2473 y Fx(a)2310 2511 y FD(\013)2368 2525 y Fx(i)2396 2511 y FD(t)p FK(\))p FD(;)125 2752 y FK(w)m(e)32 b(reac)m(h)g(the)g (equation)f FD(L)1093 2719 y Fs(00)1157 2752 y FK(+)21 b(\()p FD(n)g FC(\000)f FD(r)s FK(\))p FD(L)27 b FK(=)h FD(nL)20 b FC(\000)1947 2678 y Ft(\000)2029 2711 y Fx(n)1989 2783 y(r)r Fs(\000)p FJ(1)2113 2678 y Ft(\001)2154 2752 y FK(+)31 b(the)h FD(t)2447 2719 y Fx(r)2485 2752 y FK(-co)s(e\016cien) m(t)h(of)f(the)g(expression)139 2887 y FJ(1)p 135 2902 43 4 v 135 2954 a Fx(n)245 2836 y(n)219 2854 y Ft(P)202 3006 y Fx(k)r FJ(=1)388 2836 y Fx(n)361 2854 y Ft(P)346 3003 y Fx(p)p FJ(=1)487 2922 y FD(\020)534 2889 y FJ(2)p Fx(ak)649 2922 y FD(\013)707 2889 y FJ(2)p Fx(k)707 2945 y(p)785 2922 y FD(t)818 2889 y FJ(2)909 2836 y Fx(n)887 2854 y Ft(Q)873 3004 y Fx(i)p FJ(=1)987 2922 y FK(\(1)21 b(+)f FD(\020)1226 2889 y Fx(ak)1306 2922 y FD(\013)1364 2889 y Fx(k)1364 2948 y(i)1407 2922 y FD(t)p FK(\).)41 b(This)28 b(co)s(e\016cien)m(t)j(is)1020 3078 y Ft(\022)1160 3145 y FD(n)1087 3268 y(r)23 b FC(\000)d FK(2)1287 3078 y Ft(\023)1375 3206 y FK(+)1503 3120 y Ft(X)1465 3321 y FJ(2)p Fs(j)p Fx(n;)p FJ(2)p Fs(j)p Fx(r)1671 3206 y FK(\()p FC(\000)p FK(1\))1857 3169 y Fx(r)r(=)p FJ(2)p Fs(\000)p FJ(1)2057 3078 y Ft(\022)2197 3145 y FD(n=)p FK(2)2124 3268 y FD(r)s(=)p FK(2)h FC(\000)e FK(1)2415 3078 y Ft(\023)2482 3206 y FD( )2544 3169 y Fx(ar)r(=)p FJ(2)2690 3206 y FD(;)125 3464 y FK(where)31 b(the)h(sum)f(means)h (that)g(there)g(is)f(only)g(one)h(term)g(that)h(app)s(ears)e(when)g(2)p FC(j)p FD(n)h FK(and)f(2)p FC(j)p FD(r)s FK(.)46 b(Sub-)125 3572 y(stituting)29 b(the)h(v)-5 b(alue)30 b(of)h FD(L)f FK(and)g(simplifying,)c(w)m(e)31 b(get)125 3970 y(\(34\))562 3970 y SDict begin H.S end 562 3970 a 562 3970 a SDict begin 13 H.A end 562 3970 a 562 3970 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.34) cvn /DEST pdfmark end 562 3970 a -199 x FC(j)p FD(N)670 3733 y Fx(\031)713 3710 y Fr(00)660 3793 y Fx(n;r)760 3771 y FK(\()p FD($)s FK(\))p FC(j)26 b FK(=)f FD(L)1117 3733 y Fs(00)1185 3771 y FK(=)1271 3643 y Ft(\022)1338 3709 y FD(n)20 b FC(\000)f FK(1)1343 3833 y FD(r)k FC(\000)d FK(3)1549 3643 y Ft(\023)1636 3771 y FK(+)1763 3684 y Ft(X)1737 3873 y Fq(e)p Fr(j)p Fq(n;e)p Fr(j)p Fq(r)1775 3930 y(e)p Fr(\025)p FG(3)1931 3771 y FK(\()p FC(\000)p FK(1\))2117 3733 y Fx(r)r FJ(+)p Fx(r)r(=e)2313 3643 y Ft(\022)2380 3709 y FD(n=e)g FC(\000)g FK(1)2385 3833 y FD(r)s(=e)h FC(\000)f FK(1)2678 3643 y Ft(\023)2775 3684 y(X)2761 3882 y Fx(s)p Fs(2)p Fl(Z)2891 3863 y Fr(\003)2891 3898 y Fq(e)2936 3771 y FD( )2998 3733 y Fx(asr)r(=e)1276 4116 y FC(\000)1404 4030 y Ft(X)1367 4231 y FJ(2)p Fs(j)p Fx(n;)p FJ(2)p Fs(j)p Fx(r)1573 4116 y FK(\()p FC(\000)p FK(1\))1759 4078 y Fx(r)r(=)p FJ(2)1868 3988 y Ft(\022)1935 4054 y FD(n=)p FK(2)h FC(\000)f FK(1)1941 4178 y FD(r)s(=)p FK(2)g FC(\000)g FK(2)2237 3988 y Ft(\023)2304 4116 y FD( )2366 4078 y Fx(ar)r(=)p FJ(2)2528 4116 y FD(:)125 4317 y SDict begin H.S end 125 4317 a 125 4317 a SDict begin 13 H.A end 125 4317 a 125 4317 a SDict begin [/View [/XYZ H.V]/Dest (subsection.6.4) cvn /DEST pdfmark end 125 4317 a 107 x FK(6.4.)46 b FL(Computation)29 b(of)h(the)g(n)m(um)m(b)s(er)g(of)g(comp)s(onen)m (ts.)46 b FK(With)25 b(the)i(results)e(obtained)g(in)g(the)125 4532 y(sections)30 b(ab)s(o)m(v)m(e,)i(w)m(e)f(can)f(compute)h FD(c)p FK(\()p FD(\031)1542 4546 y FJ(1)1582 4532 y FD(;)15 b(\031)1674 4546 y FJ(2)1714 4532 y FK(\))25 b(=)g FC(j)p FD(M)1993 4487 y Fx(\031)2034 4496 y FG(1)2069 4487 y Fx(;\031)2130 4496 y FG(2)1983 4543 y Fx(n;m;r)2167 4532 y FC(j)p FK(,)31 b(for)f(gcd)q(\()p FD(n;)15 b(m)p FK(\))26 b(=)e(1.)224 4675 y(W)-8 b(e)32 b(start)f(with)e(the)h(case)i(\()p FD(\031)s(;)15 b(\031)s FK(\),)32 b(where)d(w)m(e)i(ha)m(v)m(e)557 4912 y FC(j)p FD(M)680 4875 y Fx(\031)r(;\031)670 4935 y(n;m;r)853 4912 y FC(j)25 b FK(=)1003 4799 y Fx(r)r Fs(\000)p FJ(1)999 4826 y Ft(X)1001 5020 y Fx(a)p FJ(=0)1146 4912 y FC(j)p FD(N)1254 4875 y Fx(\031)1244 4935 y(n;r)1344 4912 y FK(\()p FD( )1441 4875 y Fx(a)1484 4912 y FK(\))p FC(j)15 b(j)p FD(N)1667 4875 y Fx(\031)1657 4935 y(m;r)1778 4912 y FK(\()p FD( )1875 4875 y Fx(a)1917 4912 y FK(\))p FC(j)903 5230 y FK(=)1050 5144 y Ft(X)1023 5333 y Fq(e)p Fr(j)p Fq(n;e)p Fr(j)p Fq(r)1009 5391 y(f)6 b Fr(j)p Fq(m;f)g Fr(j)p Fq(r)1257 5169 y FK(\()p FC(\000)p FK(1\))1443 5136 y Fx(r)r(=e)p FJ(+)p Fx(r)r(=f)p 1257 5209 459 4 v 1419 5293 a FD(nm)1725 5102 y Ft(\022)1792 5169 y FD(n=e)1798 5293 y(r)s(=e)1934 5102 y Ft(\023\022)2068 5169 y FD(m=f)2086 5293 y(r)s(=f)2248 5102 y Ft(\023)2345 5144 y(X)2330 5341 y Fx(k)2367 5350 y FG(1)2401 5341 y Fx(;k)2458 5350 y FG(2)2511 5117 y Fx(r)r Fs(\000)p FJ(1)2507 5144 y Ft(X)2509 5338 y Fx(a)p FJ(=0)2654 5230 y FD(e)2696 5177 y FJ(2)p Fx(\031)r(ia)2835 5104 y Fk(\020)2888 5141 y Fq(k)2921 5156 y FG(1)p 2888 5162 68 3 v 2907 5203 a Fq(e)2965 5177 y FJ(+)3030 5141 y Fq(k)3063 5156 y FG(2)p 3030 5162 V 3046 5203 a Fq(f)3108 5104 y Fk(\021)3154 5230 y FD(;)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 23 23 TeXDict begin HPSdict begin 23 22 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.23) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(23)p 0 TeXcolorgray 125 312 a FK(where)19 b FD(k)424 326 y FJ(1)489 312 y FC(2)25 b Fz(Z)640 279 y Fs(\003)640 334 y Fx(e)675 312 y FK(,)e FD(k)770 326 y FJ(2)835 312 y FC(2)i Fz(Z)986 279 y Fs(\003)986 340 y Fx(f)1027 312 y FK(.)37 b(No)m(w)21 b FD(e)1328 259 y FJ(2)p Fx(\031)r(i)1430 186 y Fk(\020)1482 223 y Fq(k)1515 238 y FG(1)p 1482 244 68 3 v 1502 285 a Fq(e)1560 259 y FJ(+)1625 223 y Fq(k)1658 238 y FG(2)p 1625 244 V 1641 285 a Fq(f)1702 186 y Fk(\021)1774 312 y FK(=)k FD(e)1912 279 y FJ(2)p Fx(\031)r(i)p FJ(\()p Fx(k)2078 288 y FG(1)2113 279 y Fx(f)7 b FJ(+)p Fx(k)2246 288 y FG(2)2281 279 y Fx(e)p FJ(\))p Fx(=ef)2454 312 y FK(.)37 b(As)21 b FD(e)p FC(j)p FD(n;)15 b(f)10 b FC(j)p FD(m)20 b FK(and)f(gcd)q(\()p FD(n;)c(m)p FK(\))26 b(=)125 429 y(1,)35 b(w)m(e)f(ha)m(v)m(e)g(gcd)q(\()p FD(e;)15 b(f)10 b FK(\))31 b(=)f(1.)51 b(Hence)34 b Fz(Z)1514 396 y Fs(\003)1514 452 y Fx(e)1572 429 y FC(\002)22 b Fz(Z)1730 396 y Fs(\003)1730 457 y Fx(f)1801 404 y FC(\030)1801 433 y FK(=)1903 429 y Fz(Z)1968 396 y Fs(\003)1968 457 y Fx(ef)2041 429 y FK(,)35 b(via)e(the)h(map)f(\()p FD(k)2700 443 y FJ(1)2740 429 y FD(;)15 b(k)2827 443 y FJ(2)2867 429 y FK(\))31 b FC(7!)g FD(k)3102 443 y FJ(1)3142 429 y FD(f)g FK(+)22 b FD(k)3358 443 y FJ(2)3398 429 y FD(e)p FK(.)50 b(So)125 541 y(the)30 b(sum)g(only)f(con)m(tributes)h(for)g FD(e)c FK(=)f FD(f)34 b FK(=)25 b(1,)31 b(and)f(it)g(giv)m(es)125 749 y(\(35\))883 749 y SDict begin H.S end 883 749 a 883 749 a SDict begin 13 H.A end 883 749 a 883 749 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.35) cvn /DEST pdfmark end 883 749 a FC(j)p FD(M)1006 711 y Fx(\031)r(;\031)996 771 y(n;m;r)1179 749 y FC(j)c FK(=)1380 687 y(1)p 1336 728 135 4 v 1336 811 a FD(nm)1480 621 y Ft(\022)1547 687 y FD(n)1552 811 y(r)1601 621 y Ft(\023\022)1735 687 y FD(m)1753 811 y(r)1815 621 y Ft(\023)1882 749 y FD(r)i FK(=)2057 687 y(1)p 2057 728 46 4 v 2058 811 a FD(r)2112 621 y Ft(\022)2179 687 y FD(n)20 b FC(\000)g FK(1)2185 811 y FD(r)i FC(\000)e FK(1)2390 621 y Ft(\023\022)2524 687 y FD(m)g FC(\000)g FK(1)2542 811 y FD(r)j FC(\000)d FK(1)2760 621 y Ft(\023)2827 749 y FD(:)224 997 y FK(This)26 b(form)m(ula)i(agrees)h(with)d([)1261 997 y SDict begin H.S end 1261 997 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 1353 938 a SDict begin H.R end 1353 938 a 1353 997 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 1353 997 a FK(,)i(Thm.)f(6.1],)k(whic)m(h)26 b(sa)m(ys)j(that)f(the)h(c)m(haracter)g(v)-5 b(ariet)m(y)29 b FC(R)3414 1011 y Fx(r)3480 997 y FK(has)125 1105 y(dimension)d(\()p FD(r)21 b FC(\000)d FK(1\))819 1072 y FJ(2)860 1105 y FK(,)29 b(and)g(the)h(n)m(um)m(b)s(er)e(of)h(irreducible)d(comp)s(onen) m(ts)k(of)f(this)f(dimension)f(is)i(\()3460 1105 y SDict begin H.S end 3460 1105 a 0 TeXcolorgray FK(35)p 0 TeXcolorgray 3551 1046 a SDict begin H.R end 3551 1046 a 3551 1105 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.35) cvn H.B /ANN pdfmark end 3551 1105 a FK(\).)125 1273 y SDict begin H.S end 125 1273 a 125 1273 a SDict begin 13 H.A end 125 1273 a 125 1273 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.3) cvn /DEST pdfmark end 125 1273 a 0 TeXcolorgray FL(Remark)38 b(6.3.)p 0 TeXcolorgray 44 w FK(W)-8 b(e)35 b(can)f(get)h(a)f(form)m(ula)f(when)g(gcd)q(\()p FD(n;)15 b(m)p FK(\))31 b FD(>)g FK(1)j(as)g(w)m(ell.)50 b(W)-8 b(e)35 b(will)c(not)j(dev)m(elop)125 1381 y(this.)224 1586 y(F)-8 b(or)31 b(the)g(case)g(\()p FD(\031)s(;)15 b(\031)921 1553 y Fs(0)945 1586 y FK(\),)31 b(w)m(e)g(use)f(\()1362 1586 y SDict begin H.S end 1362 1586 a 0 TeXcolorgray FK(33)p 0 TeXcolorgray 1454 1527 a SDict begin H.R end 1454 1527 a 1454 1586 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.33) cvn H.B /ANN pdfmark end 1454 1586 a FK(\))g(whic)m(h)f(sa)m (ys)i(that)g FC(j)p FD(N)2277 1553 y Fx(\031)2320 1529 y Fr(0)2267 1608 y Fx(m;r)2388 1586 y FC(j)25 b FK(=)g FC(\000)p FD(r)s FC(j)p FD(N)2757 1553 y Fx(\031)2747 1608 y(m;r)2866 1586 y FC(j)c FK(+)3003 1512 y Ft(\000)3075 1545 y Fx(m)3045 1617 y(r)r Fs(\000)p FJ(1)3169 1512 y Ft(\001)3210 1586 y FK(,)31 b(hence)364 1847 y FC(j)p FD(M)487 1810 y Fx(\031)r(;\031)593 1787 y Fr(0)477 1870 y Fx(n;m;r)660 1847 y FC(j)25 b FK(=)810 1734 y Fx(r)r Fs(\000)p FJ(1)806 1761 y Ft(X)808 1955 y Fx(a)p FJ(=0)953 1847 y FC(j)p FD(N)1061 1810 y Fx(\031)1051 1870 y(n;r)1151 1847 y FK(\()p FD( )1248 1810 y Fx(a)1291 1847 y FK(\))p FC(j)15 b(j)p FD(N)1474 1810 y Fx(\031)1517 1787 y Fr(0)1464 1870 y Fx(m;r)1585 1847 y FK(\()p FD( )1682 1810 y Fx(a)1724 1847 y FK(\))p FC(j)26 b FK(=)f FC(\000)p FD(r)s FC(j)p FD(M)2144 1810 y Fx(\031)r(;\031)2134 1870 y(n;m;r)2316 1847 y FC(j)c FK(+)2453 1719 y Ft(\022)2580 1786 y FD(m)2519 1910 y(r)i FC(\000)d FK(1)2720 1719 y Ft(\023)2805 1734 y Fx(r)r Fs(\000)p FJ(1)2802 1761 y Ft(X)2803 1955 y Fx(a)p FJ(=0)2948 1847 y FC(j)p FD(N)3056 1810 y Fx(\031)3046 1870 y(n;r)3147 1847 y FK(\()p FD( )3244 1810 y Fx(a)3286 1847 y FK(\))p FC(j)p FD(:)125 2161 y FK(No)m(w)31 b(use)f(\()524 2161 y SDict begin H.S end 524 2161 a 0 TeXcolorgray FK(30)p 0 TeXcolorgray 615 2102 a SDict begin H.R end 615 2102 a 615 2161 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.30) cvn H.B /ANN pdfmark end 615 2161 a FK(\))h(and)859 2075 y Fx(r)r Fs(\000)p FJ(1)873 2093 y Ft(P)857 2242 y Fx(a)p FJ(=0)1034 2093 y Ft(P)1000 2241 y Fx(\027)t Fs(2)p Fx(\026)1128 2222 y Fr(\003)1128 2257 y Fq(e)1180 2161 y FD(\027)1231 2128 y Fx(a)1297 2161 y FK(=)1428 2093 y Ft(P)1393 2241 y Fx(\027)t Fs(2)p Fx(\026)1521 2222 y Fr(\003)1521 2257 y Fq(e)1575 2075 y Fx(r)r Fs(\000)p FJ(1)1589 2093 y Ft(P)1573 2242 y Fx(a)p FJ(=0)1716 2161 y FD(\027)1767 2128 y Fx(a)1833 2161 y FK(=)25 b(0)31 b(for)f FD(e)c(>)f FK(1.)41 b(Then)125 2456 y(\(36\))469 2456 y SDict begin H.S end 469 2456 a 469 2456 a SDict begin 13 H.A end 469 2456 a 469 2456 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.36) cvn /DEST pdfmark end 469 2456 a FC(j)p FD(M)592 2419 y Fx(\031)r(;\031)698 2396 y Fr(0)582 2479 y Fx(n;m;r)765 2456 y FC(j)25 b FK(=)g FC(\000)982 2328 y Ft(\022)1049 2395 y FD(n)20 b FC(\000)g FK(1)1055 2519 y FD(r)i FC(\000)e FK(1)1260 2328 y Ft(\023\022)1394 2395 y FD(m)g FC(\000)g FK(1)1412 2519 y FD(r)j FC(\000)d FK(1)1630 2328 y Ft(\023)1717 2456 y FK(+)1808 2328 y Ft(\022)1935 2395 y FD(m)1875 2519 y(r)j FC(\000)d FK(1)2075 2328 y Ft(\023)2158 2395 y FD(r)p 2152 2436 55 4 v 2152 2519 a(n)2217 2328 y Ft(\022)2284 2395 y FD(n)2289 2519 y(r)2338 2328 y Ft(\023)2431 2456 y FK(=)2526 2328 y Ft(\022)2593 2395 y FD(n)g FC(\000)g FK(1)2599 2519 y FD(r)j FC(\000)d FK(1)2805 2328 y Ft(\023)o(\022)2938 2395 y FD(m)g FC(\000)g FK(1)2956 2519 y FD(r)j FC(\000)d FK(2)3175 2328 y Ft(\023)3242 2456 y FD(:)224 2706 y FK(A)31 b(similar)c(argumen)m(t)k(sho)m(ws)f(the)h(case)g(\()p FD(\031)1728 2673 y Fs(0)1752 2706 y FD(;)15 b(\031)1847 2673 y Fs(0)1871 2706 y FK(\),)125 3046 y(\(37\))496 3046 y SDict begin H.S end 496 3046 a 496 3046 a SDict begin 13 H.A end 496 3046 a 496 3046 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.37) cvn /DEST pdfmark end 496 3046 a -127 x FC(j)p FD(M)619 2881 y Fx(\031)662 2858 y Fr(0)684 2881 y Fx(;\031)747 2858 y Fr(0)609 2941 y Fx(n;m;r)792 2919 y FC(j)25 b FK(=)15 b FD(r)972 2791 y Ft(\022)1038 2857 y FD(n)20 b FC(\000)g FK(1)1044 2981 y FD(r)j FC(\000)d FK(1)1250 2791 y Ft(\023\022)1383 2857 y FD(m)g FC(\000)g FK(1)1402 2981 y FD(r)i FC(\000)e FK(1)1620 2791 y Ft(\023)1707 2919 y FC(\000)g FD(r)1842 2791 y Ft(\022)1968 2857 y FD(m)1908 2981 y(r)j FC(\000)d FK(1)2108 2791 y Ft(\023)2191 2857 y FD(r)p 2185 2898 55 4 v 2185 2981 a(n)2250 2791 y Ft(\022)2317 2857 y FD(n)2322 2981 y(r)2371 2791 y Ft(\023)2458 2919 y FC(\000)g FD(r)2593 2791 y Ft(\022)2733 2857 y FD(n)2660 2981 y(r)j FC(\000)c FK(1)2860 2791 y Ft(\023)2955 2857 y FD(r)p 2937 2898 80 4 v 2937 2981 a(m)3027 2791 y Ft(\022)3093 2857 y FD(m)3112 2981 y(r)3173 2791 y Ft(\023)948 3172 y FK(+)h FD(r)1083 3044 y Ft(\022)1222 3111 y FD(n)1150 3235 y(r)i FC(\000)e FK(1)1350 3044 y Ft(\023\022)1544 3111 y FD(m)1484 3235 y(r)i FC(\000)e FK(1)1684 3044 y Ft(\023)1776 3172 y FK(=)25 b FD(r)1916 3044 y Ft(\022)1982 3111 y FD(n)20 b FC(\000)g FK(1)1988 3235 y FD(r)i FC(\000)e FK(2)2193 3044 y Ft(\023\022)2327 3111 y FD(m)g FC(\000)g FK(1)2345 3235 y FD(r)j FC(\000)d FK(2)2564 3044 y Ft(\023)2630 3172 y FD(:)224 3422 y FK(Regarding)33 b(the)g(partition)f FD(\031)1259 3389 y Fs(00)1301 3422 y FK(,)i(w)m(e)g(only)e(write)g (the)i(form)m(ula)e(for)h(\()p FD(\031)s(;)15 b(\031)2757 3389 y Fs(00)2800 3422 y FK(\),)34 b(as)g(this)e(is)g(the)h(only)125 3530 y(case)e(that)g(w)m(e)g(need)f(for)g FD(r)e FK(=)d(4.)41 b(Using)29 b(expression)h(\()2004 3530 y SDict begin H.S end 2004 3530 a 0 TeXcolorgray FK(34)p 0 TeXcolorgray 2095 3472 a SDict begin H.R end 2095 3472 a 2095 3530 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.34) cvn H.B /ANN pdfmark end 2095 3530 a FK(\))g(as)h(b)s(efore)f(w)m(e)h(get)125 3743 y(\(38\))1277 3743 y SDict begin H.S end 1277 3743 a 1277 3743 a SDict begin 13 H.A end 1277 3743 a 1277 3743 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.38) cvn /DEST pdfmark end 1277 3743 a FC(j)p FD(M)1400 3705 y Fx(\031)r(;\031)1506 3682 y Fr(00)1390 3765 y Fx(n;m;r)1572 3743 y FC(j)26 b FK(=)1719 3615 y Ft(\022)1786 3681 y FD(n)20 b FC(\000)g FK(1)1791 3805 y FD(r)j FC(\000)d FK(1)1997 3615 y Ft(\023\022)2131 3681 y FD(m)g FC(\000)g FK(1)2149 3805 y FD(r)j FC(\000)d FK(3)2367 3615 y Ft(\023)2434 3743 y FD(:)224 3998 y FK(F)-8 b(or)44 b(rank)e FD(r)48 b FK(=)d(4,)i(the)42 b(only)g(allo)m(w)m(able)g(partitions)f(are)i FD(\031)49 b FK(=)d FC(f)p FK(1)2629 3965 y FJ(4)2669 3998 y FC(g)p FK(,)g FD(\031)2840 3965 y Fs(0)2909 3998 y FK(=)g FC(f)p FK(1)3116 3965 y FJ(2)3156 3998 y FD(;)15 b FK(2)3241 3965 y FJ(1)3281 3998 y FC(g)p FK(,)47 b FD(\031)3453 3965 y Fs(00)3541 3998 y FK(=)125 4106 y FC(f)p FK(3)215 4073 y FJ(1)255 4106 y FD(;)15 b FK(1)340 4073 y FJ(1)380 4106 y FC(g)p FK(,)48 b FD(\031)553 4073 y Fs(000)662 4106 y FK(=)f FC(f)p FK(2)870 4073 y FJ(2)910 4106 y FC(g)p FK(.)81 b(All)43 b(cases)h(ha)m(v)m(e)h(b)s(een)e(giv)m (en)h(except)h FD(\031)2521 4073 y Fs(000)2630 4106 y FK(=)i FC(f)p FK(2)2838 4073 y FJ(2)2878 4106 y FC(g)p FK(.)81 b(F)-8 b(or)45 b(suc)m(h)e(case,)125 4214 y FD(\026)180 4228 y FJ(4)244 4214 y FK(=)25 b FC(f)p FK(1)p FD(;)15 b FC(\000)p FK(1)p FD(;)g(i;)g FC(\000)p FD(i)p FC(g)p FK(,)34 b(and)777 4399 y FD(N)860 4351 y Fs(f)p FJ(2)930 4327 y FG(2)965 4351 y Fs(g)850 4425 y Fx(n;)p FJ(4)1004 4399 y FK(\()p FD($)s FK(\))26 b(=)f FC(f)p FK(\()p FD(\017)1391 4413 y FJ(1)1431 4399 y FD(;)15 b(\017)1508 4413 y FJ(2)1548 4399 y FK(\))p FC(j)p FD(\017)1645 4413 y FJ(1)1710 4399 y FC(6)p FK(=)25 b FD(\017)1843 4413 y FJ(2)1882 4399 y FD(;)15 b(\017)1959 4361 y Fx(n)1959 4421 y FJ(1)2032 4399 y FK(=)25 b FD($)s(;)15 b(\017)2283 4361 y Fx(n)2283 4421 y FJ(2)2355 4399 y FK(=)25 b FD($)s(;)15 b(\017)2606 4361 y FJ(2)2606 4421 y(1)2646 4399 y FD(\017)2683 4361 y FJ(2)2683 4421 y(2)2747 4399 y FK(=)25 b(1)p FC(g)p FD(:)125 4571 y FK(The)k(map)h(\()p FD(\017)585 4585 y FJ(1)625 4571 y FD(;)15 b(\017)702 4585 y FJ(2)742 4571 y FK(\))25 b FC(7!)h FK(\()p FD(\017)991 4538 y FJ(2)991 4596 y(1)1030 4571 y FD(;)15 b(\017)1107 4538 y FJ(2)1107 4596 y(2)1147 4571 y FK(\))31 b(giv)m(es)f(that)1161 4757 y FC(j)p FD(N)1269 4709 y Fs(f)p FJ(2)1339 4686 y FG(2)1374 4709 y Fs(g)1259 4783 y Fx(n;)p FJ(4)1414 4757 y FK(\()p FD($)s FK(\))p FC(j)c FK(=)f FC(j)p FD(N)1817 4709 y Fs(f)p FJ(1)1887 4686 y FG(2)1922 4709 y Fs(g)1807 4783 y Fx(n;)p FJ(2)1962 4757 y FK(\()p FD($)2075 4720 y FJ(2)2114 4757 y FK(\))p FC(j)c FK(+)f FD(A)2354 4771 y Fx(n)2401 4757 y FK(\()p FD($)s FK(\))p FD(;)125 4930 y FK(where)25 b FD(A)451 4944 y Fx(n)499 4930 y FK(\()p FD($)s FK(\))h(coun)m(ts)h(the)g(n)m(um)m(b)s(er)e(of)h FD(\017)1571 4944 y FJ(1)1636 4930 y FC(6)p FK(=)f FD(\017)1769 4944 y FJ(2)1834 4930 y FK(with)g FD(\017)2074 4897 y FJ(2)2074 4955 y(1)2139 4930 y FK(=)g FD(\017)2272 4897 y FJ(2)2272 4955 y(2)2311 4930 y FK(,)j FD(\017)2401 4897 y Fx(n)2401 4955 y FJ(1)2473 4930 y FK(=)d FD(\017)2606 4897 y Fx(n)2606 4955 y FJ(2)2678 4930 y FK(=)g FD($)k FK(and)d FD(\017)3088 4897 y FJ(2)3088 4955 y(1)3127 4930 y FD(\017)3164 4897 y FJ(2)3164 4955 y(2)3228 4930 y FK(=)f(1.)40 b(This)125 5038 y(means)30 b(that)h(\()p FD(\017)672 5052 y FJ(1)711 5038 y FD(;)15 b(\017)788 5052 y FJ(2)828 5038 y FK(\))26 b(=)f(\(1)p FD(;)15 b FC(\000)p FK(1\))p FD(;)g FK(\()p FC(\000)p FK(1)p FD(;)g FK(1\))p FD(;)g FK(\()p FD(i;)g FC(\000)p FD(i)p FK(\))p FD(;)g FK(\()p FC(\000)p FD(i;)h(i)p FK(\).)47 b(Then)481 5305 y FD(A)549 5319 y Fx(n)596 5305 y FK(\()p FD($)s FK(\))26 b(=)866 5119 y Ft(8)866 5201 y(<)866 5364 y(:)988 5196 y FK(2)p FD(;)265 b(n)25 b FC(\021)g FK(2)92 b(\(mo)s(d)29 b(4\))p FD(;)j($)c FK(=)d FC(\006)p FK(1)988 5304 y(4)p FD(;)265 b(n)25 b FC(\021)g FK(0)92 b(\(mo)s(d)29 b(4\))p FD(;)j($)c FK(=)d(1)988 5412 y(0)p FD(;)265 b FK(otherwise)2368 5119 y Ft(9)2368 5201 y(=)2368 5364 y(;)2475 5305 y FK(=)2606 5219 y Ft(X)2571 5420 y Fx(e)p Fs(j)p Fx(n;e)p Fs(j)p FJ(4)2802 5219 y Ft(X)2788 5416 y Fx(s)p Fs(2)p Fl(Z)2918 5397 y Fr(\003)2918 5433 y Fq(e)2963 5305 y FD( )3025 5268 y Fx(as)12 b FJ(4)p Fx(=e)3230 5305 y FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.24) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(24)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 224 278 a FK(W)-8 b(e)31 b(use)f(this)e(form)m(ula)h(to)i(compute)f FC(j)p FD(M)1648 234 y Fx(\031)1689 243 y FG(1)1723 234 y Fx(;\031)1786 210 y Fr(000)1638 304 y Fx(n;m;)p FJ(4)1849 278 y FC(j)p FK(,)g(for)f FD(\031)2119 292 y FJ(1)2184 278 y FK(=)c FD(\031)s(;)15 b(\031)2430 245 y Fs(0)2454 278 y FK(.)40 b(Note)31 b(that)g(the)f(case)g FD(\031)3334 292 y FJ(1)3399 278 y FK(=)25 b FD(\031)3550 245 y Fs(000)125 388 y FK(is)k(not)i(needed)f(due)f(to)i(Prop)s(osition)1453 388 y SDict begin H.S end 1453 388 a 0 TeXcolorgray FK(8.1)p 0 TeXcolorgray 1569 330 a SDict begin H.R end 1569 330 a 1569 388 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.8.1) cvn H.B /ANN pdfmark end 1569 388 a FK(.)41 b(After)30 b(some)h(manipulations,)125 714 y(\(39\))1266 714 y SDict begin H.S end 1266 714 a 1266 714 a SDict begin 13 H.A end 1266 714 a 1266 714 a SDict begin [/View [/XYZ H.V]/Dest (equation.6.39) cvn /DEST pdfmark end 1266 714 a 1283 587 a FC(j)p FD(M)1406 543 y Fx(\031)r(;\031)1512 520 y Fr(000)1396 613 y Fx(n;m;)p FJ(4)1580 587 y FC(j)26 b FK(=)1737 526 y(1)p 1737 566 46 4 v 1737 650 a(2)1792 459 y Ft(\022)1859 526 y FD(n)20 b FC(\000)g FK(1)1942 650 y(3)2070 459 y Ft(\023)2137 587 y FK(\()p FD(m)h FC(\000)f FK(1\))p FD(;)1266 841 y FC(j)p FD(M)1389 797 y Fx(\031)1432 773 y Fr(0)1455 797 y Fx(;\031)1518 773 y Fr(000)1379 867 y Fx(n;m;)p FJ(4)1580 841 y FC(j)26 b FK(=)f(2)1772 713 y Ft(\022)1839 779 y FD(n)20 b FC(\000)g FK(1)1922 903 y(2)2050 713 y Ft(\023)2117 841 y FK(\()p FD(m)h FC(\000)f FK(1\))p FD(:)125 988 y SDict begin H.S end 125 988 a 125 988 a SDict begin 13 H.A end 125 988 a 125 988 a SDict begin [/View [/XYZ H.V]/Dest (subsection.6.5) cvn /DEST pdfmark end 125 988 a 107 x FK(6.5.)46 b FL(Case)j(of)h(coprime)g(indices.)c FK(In)c(this)g(section)i(w)m(e)f (tak)m(e)i(a)e(di\013eren)m(t)g(approac)m(h)g(to)h(the)125 1203 y(computation)23 b(of)h(the)g(n)m(um)m(b)s(er)e(of)i(comp)s(onen)m (ts.)38 b(This)22 b(approac)m(h)i(will)d(b)s(e)i(v)-5 b(alid)22 b(for)h(an)m(y)h(partitions)125 1311 y FD(\031)177 1325 y FJ(1)216 1311 y FD(;)15 b(\031)308 1325 y FJ(2)381 1311 y FK(of)33 b FD(r)j FK(but,)d(on)g(the)h(other)f(hand,)g(it)g (will)d(only)j(w)m(ork)g(when)f(w)m(e)i(add)e(the)i(extra)g(h)m(yp)s (othesis)125 1419 y(gcd\()p FD(n;)15 b(r)s FK(\))25 b(=)g(gcd)q(\()p FD(m;)15 b(r)s FK(\))25 b(=)g(1.)224 1568 y(Consider)32 b(the)h(group)g(isomorphism)d Fz(Z)1626 1582 y Fx(s)1689 1568 y FC(!)g FD(\026)1865 1582 y Fx(s)1901 1568 y FK(,)35 b FD(k)e FC(7!)d FD(e)2204 1535 y FJ(2)p Fx(\031)r(ik)r(=s)2417 1568 y FK(.)49 b(Under)33 b(this)f(isomorphism,)f(let)125 1677 y FD($)d FK(=)d FD(e)366 1644 y FJ(2)p Fx(\031)r(ik)r(=r)610 1677 y FK(for)30 b(some)h FD(k)e FC(2)24 b Fz(Z)1203 1691 y Fx(r)1237 1677 y FK(.)41 b(In)m(tro)s(duce)29 b(the)i(sets)347 1803 y(~)321 1826 y FD(N)394 1840 y Fx(n;r)495 1826 y FK(\()p FD(k)s(;)15 b(l)r FK(\))26 b(=)f FC(f)p FK(\()p FD(x)938 1840 y FJ(1)978 1826 y FD(;)15 b(:)g(:)g(:)i(;)e(x)1232 1840 y Fx(r)1270 1826 y FK(\))26 b FC(2)f Fz(Z)1482 1788 y Fx(r)1482 1848 y(nr)1573 1826 y FC(j)15 b FD(nx)1720 1840 y Fx(i)1773 1826 y FC(\021)25 b FD(k)34 b FK(\(mo)s(d)c FD(r)s FK(\))p FD(;)15 b(x)2361 1840 y FJ(1)2421 1826 y FK(+)20 b FC(\001)15 b(\001)g(\001)21 b FK(+)f FD(x)2781 1840 y Fx(r)2844 1826 y FC(\021)25 b FD(l)32 b FK(\(mo)s(d)e FD(n)p FK(\))p FC(g)16 b FD(:)125 1988 y FK(Then)29 b(the)h(set)687 1965 y(^)660 1988 y FD(N)733 2002 y Fx(n;r)834 1988 y FK(\()p FD($)s FK(\))h(in)e(\()1154 1988 y SDict begin H.S end 1154 1988 a 0 TeXcolorgray FK(26)p 0 TeXcolorgray 1245 1929 a SDict begin H.R end 1245 1929 a 1245 1988 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.26) cvn H.B /ANN pdfmark end 1245 1988 a FK(\))i(has)f(the)g (same)h(cardinalit)m(y)e(as)2450 1965 y(~)2423 1988 y FD(N)2496 2002 y Fx(n;r)2597 1988 y FK(\()p FD(k)s(;)15 b FK(0\).)125 2152 y SDict begin H.S end 125 2152 a 125 2152 a SDict begin 13 H.A end 125 2152 a 125 2152 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.4) cvn /DEST pdfmark end 125 2152 a 0 TeXcolorgray FL(Prop)s(osition)36 b(6.4.)p 0 TeXcolorgray 42 w Fp(If)c FK(gcd)q(\()p FD(n;)15 b(r)s FK(\))25 b(=)g(1)33 b Fp(then)795 2308 y FC(j)847 2285 y FK(~)820 2308 y FD(N)893 2322 y Fx(n;r)994 2308 y FK(\()p FD(k)s(;)15 b(l)r FK(\))p FC(j)27 b FK(=)e FC(j)15 b(f)q FK(\()p FD(\013)1510 2322 y FJ(1)1550 2308 y FD(;)g(:)g(:)g(:)h(;)f(\013)1809 2322 y Fx(r)1848 2308 y FK(\))25 b FC(2)g Fz(Z)2059 2270 y Fx(r)2059 2330 y(n)2117 2308 y FC(j)15 b FD(\013)2215 2322 y FJ(1)2275 2308 y FK(+)20 b FC(\001)15 b(\001)g(\001)21 b FK(+)f FD(\013)2641 2322 y Fx(r)2705 2308 y FK(=)25 b FD(l)r FC(g)15 b(j)p FD(;)125 2457 y Fp(for)33 b(any)g FD(k)28 b FC(2)d Fz(Z)670 2471 y Fx(r)736 2457 y Fp(and)34 b FD(l)27 b FC(2)e Fz(Z)1118 2471 y Fx(n)1161 2457 y Fp(.)p 0 TeXcolorgray 125 2654 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(Consider)30 b(the)i(homomorphism)d FD(')f FK(:)f Fz(Z)1791 2621 y Fx(r)1791 2677 y(nr)1895 2654 y FC(!)g Fz(Z)2078 2668 y Fx(n)2141 2654 y FC(\002)21 b Fz(Z)2298 2621 y Fx(r)2298 2677 y(r)2332 2654 y FK(,)32 b(giv)m(en)f(b)m(y)h FD(')p FK(\()p FD(x)2901 2668 y FJ(1)2941 2654 y FD(;)15 b(:)g(:)g(:)h(;)f(x) 3194 2668 y Fx(r)3233 2654 y FK(\))27 b(=)g(\()p FD(x)3480 2668 y FJ(1)3541 2654 y FK(+)125 2769 y FC(\001)15 b(\001)g(\001)24 b FK(+)f FD(x)400 2783 y Fx(r)469 2769 y FK(\(mo)s(d)29 b FD(n)p FK(\))p FD(;)15 b(nx)945 2783 y Fx(i)1004 2769 y FK(\(mo)s(d)30 b FD(r)s FK(\)\).)55 b(W)-8 b(e)37 b(ha)m(v)m(e)g (that)2043 2746 y(~)2017 2769 y FD(N)2090 2783 y Fx(n;r)2190 2769 y FK(\()p FD(k)s(;)15 b(l)r FK(\))35 b(=)e FD(')2577 2736 y Fs(\000)p FJ(1)2672 2769 y FK(\()p FD(l)r(;)15 b(k)s(;)g(k)s(;)g(:)g(:)g(:)j(;)d(k)s FK(\).)56 b(In)35 b(order)125 2877 y(to)c(iden)m(tify)d(this)i(map,)g(observ)m(e)h(that)g FD(')g FK(factors)g(as)1484 3080 y Fz(Z)1549 3047 y Fx(r)1549 3102 y(nr)1768 3017 y(\036)1810 2994 y Fq(r)1872 3057 y Fw(/)p Fv(/)p 1653 3059 220 4 v 1750 3282 a Fx(')1959 3329 y Fw($)p Fv($)1924 3306 y Fu(I)1890 3283 y(I)1856 3260 y(I)1822 3237 y(I)1789 3214 y(I)1755 3191 y(I)1721 3169 y(I)1687 3146 y(I)1653 3123 y(I)1899 3080 y Fz(Z)1964 3047 y Fx(r)1964 3102 y(n)2027 3080 y FC(\002)20 b Fz(Z)2183 3047 y Fx(r)2183 3102 y(r)2085 3243 y(f)7 b Fs(\002)p Fx(g)2217 3220 y Fq(r)2058 3329 y Fw(\017)p Fv(\017)p 2056 3329 4 200 v 1879 3418 a Fz(Z)1944 3432 y Fx(n)2007 3418 y FC(\002)20 b Fz(Z)2163 3385 y Fx(r)2163 3441 y(r)2211 3418 y FD(;)125 3580 y FK(where)41 b(the)i(horizon)m(tal)f(map)g(is)f (the)i(isomorphism)c FD(\036)45 b FK(:)h Fz(Z)2271 3594 y Fx(nr)2392 3555 y FC(\030)2392 3585 y FK(=)2508 3580 y Fz(Z)2573 3594 y Fx(n)2644 3580 y FC(\002)28 b Fz(Z)2808 3594 y Fx(r)2884 3580 y FK(giv)m(en)42 b(b)m(y)g(the)h(map)125 3688 y FD(x)d FC(7!)g FK(\()p FD(x)h FK(\(mo)s(d)30 b FD(n)p FK(\))p FD(;)15 b(x)41 b FK(\(mo)s(d)29 b FD(r)s FK(\)\).)69 b(F)-8 b(or)40 b(the)g(v)m(ertical)f(arro)m(w)h(w)m(e)g(ha) m(v)m(e)g FD(f)50 b FK(:)41 b Fz(Z)2908 3655 y Fx(r)2908 3711 y(n)2991 3688 y FC(!)f Fz(Z)3187 3702 y Fx(n)3269 3688 y FK(giv)m(en)f(b)m(y)125 3796 y FD(f)10 b FK(\()p FD(\013)273 3810 y FJ(1)312 3796 y FD(;)15 b(:)g(:)g(:)h(;)f(\013)571 3810 y Fx(r)610 3796 y FK(\))26 b(=)e FD(\013)824 3810 y FJ(1)884 3796 y FK(+)c FD(:)15 b(:)g(:)21 b FK(+)f FD(\013)1250 3810 y Fx(r)1288 3796 y FK(,)31 b(and)f FD(g)f FK(:)c Fz(Z)1708 3810 y Fx(r)1767 3796 y FC(!)g Fz(Z)1948 3810 y Fx(r)2012 3796 y FK(is)k FD(g)s FK(\()p FD(\014)5 b FK(\))27 b(=)e FD(n\014)5 b FK(.)41 b(Therefore,)159 3952 y FD(\036)213 3915 y Fx(r)251 3952 y FK(\()313 3929 y(~)286 3952 y FD(N)359 3966 y Fx(n;r)460 3952 y FK(\()p FD(k)s(;)15 b(l)r FK(\)\))27 b(=)e FC(f)p FK(\()p FD(\013)945 3966 y FJ(1)985 3952 y FD(;)15 b(:)g(:)g(:)i(;)e(\013)1245 3966 y Fx(r)1283 3952 y FK(\))26 b FC(2)f Fz(Z)1495 3915 y Fx(r)1495 3975 y(n)1553 3952 y FC(j)15 b FD(\013)1651 3966 y FJ(1)1711 3952 y FK(+)20 b FC(\001)15 b(\001)g(\001)21 b FK(+)f FD(\013)2077 3966 y Fx(r)2140 3952 y FK(=)25 b FD(l)r FC(g)c(\002)f(f)p FK(\()p FD(\014)2553 3966 y FJ(1)2593 3952 y FD(;)15 b(:)g(:)g(:)i(;)e(\014)2846 3966 y Fx(r)2885 3952 y FK(\))25 b FC(2)g Fz(Z)3096 3915 y Fx(r)3096 3975 y(r)3145 3952 y FC(j)15 b FD(n\014)3291 3966 y Fx(i)3345 3952 y FK(=)25 b FD(k)s FC(g)16 b FD(:)125 4101 y FK(Moreo)m(v)m(er,)32 b(since)e FD(n)g FK(is)f(in)m(v)m(ertible) g(in)g Fz(Z)1521 4115 y Fx(r)1555 4101 y FK(,)h(the)h(last)f(factor)h (is)f(just)g(a)g(p)s(oin)m(t.)40 b(Hence)446 4252 y FC(j)498 4229 y FK(~)471 4252 y FD(N)544 4266 y Fx(n;r)645 4252 y FK(\()p FD(k)s(;)15 b(l)r FK(\))p FC(j)27 b FK(=)e FC(j)p FD(\036)1061 4215 y Fx(r)1099 4252 y FK(\()1161 4229 y(~)1134 4252 y FD(N)1207 4266 y Fx(n;r)1308 4252 y FK(\()p FD(k)s(;)15 b(l)r FK(\)\))p FC(j)27 b FK(=)e FC(j)15 b(f)q FK(\()p FD(\013)1859 4266 y FJ(1)1899 4252 y FD(;)g(:)g(:)g(:)h(;)f(\013)2158 4266 y Fx(r)2197 4252 y FK(\))25 b FC(2)g Fz(Z)2408 4215 y Fx(r)2408 4275 y(n)2466 4252 y FC(j)15 b FD(\013)2564 4266 y FJ(1)2624 4252 y FK(+)20 b FC(\001)15 b(\001)g(\001)21 b FK(+)f FD(\013)2990 4266 y Fx(r)3054 4252 y FK(=)25 b FD(l)r FC(g)15 b(j)p FD(:)3541 4401 y Fo(\003)125 4565 y SDict begin H.S end 125 4565 a 125 4565 a SDict begin 13 H.A end 125 4565 a 125 4565 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.5) cvn /DEST pdfmark end 125 4565 a 0 TeXcolorgray FL(Corollary)36 b(6.5.)p 0 TeXcolorgray 42 w Fp(If)d FK(gcd\()p FD(n;)15 b(r)s FK(\))25 b(=)g(1)p Fp(,)33 b(then)g FC(j)1717 4542 y FK(~)1690 4565 y FD(N)1763 4579 y Fx(n;r)1864 4565 y FK(\()p FD(k)s(;)15 b(l)r FK(\))p FC(j)34 b Fp(is)f(indep)-5 b(endent)34 b(of)f FD(k)i Fp(and)f FD(l)r Fp(.)p 0 TeXcolorgray 125 4763 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(The)34 b(indep)s(endence)d(of)j FD(k)j FK(follo)m(ws)c(from)g(Prop)s(osition) 2366 4763 y SDict begin H.S end 2366 4763 a 0 TeXcolorgray FK(6.4)p 0 TeXcolorgray 2482 4704 a SDict begin H.R end 2482 4704 a 2482 4763 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6.4) cvn H.B /ANN pdfmark end 2482 4763 a FK(.)51 b(F)-8 b(or)34 b(the)g(indep)s(endence)d(of)j FD(l)r FK(,)125 4871 y(observ)m(e)c(that)h(the)f(map)g(\()p FD(x)1090 4885 y FJ(1)1130 4871 y FD(;)15 b(:)g(:)g(:)i(;)e(x)1384 4885 y Fx(r)1422 4871 y FK(\))26 b FC(7!)f FK(\()p FD(x)1686 4885 y FJ(1)1745 4871 y FK(+)20 b FD(r)1880 4838 y Fs(\000)p FJ(1)1974 4871 y FK(\()p FD(l)2038 4838 y Fs(0)2081 4871 y FC(\000)g FD(l)r FK(\))p FD(;)15 b(:)g(:)g(:)i(;)e(x)2490 4885 y Fx(r)2548 4871 y FK(+)k FD(r)2682 4838 y Fs(\000)p FJ(1)2776 4871 y FK(\()p FD(l)2840 4838 y Fs(0)2883 4871 y FC(\000)h FD(l)r FK(\)\))31 b(is)e(a)i(bijection)125 4985 y(b)s(et)m(w)m(een)494 4962 y(~)468 4985 y FD(N)541 4999 y Fx(n;r)641 4985 y FK(\(0)p FD(;)15 b(l)r FK(\))24 b(and)1045 4962 y(~)1018 4985 y FD(N)1091 4999 y Fx(n;r)1192 4985 y FK(\(0)p FD(;)15 b(l)1341 4952 y Fs(0)1365 4985 y FK(\),)25 b(for)d(an)m(y)h FD(l)r(;)15 b(l)1843 4952 y Fs(0)1892 4985 y FC(2)25 b Fz(Z)2043 4999 y Fx(n)2086 4985 y FK(,)f(where)e FD(r)2434 4952 y Fs(\000)p FJ(1)2551 4985 y FK(is)f(the)i(in)m(v)m(erse)g(of)f FD(r)43 b FK(\(mo)s(d)30 b FD(n)p FK(\),)125 5093 y(and)f(lifted)g(to)i Fz(Z)712 5107 y Fx(nr)789 5093 y FK(.)2727 b Fo(\003)125 5257 y SDict begin H.S end 125 5257 a 125 5257 a SDict begin 13 H.A end 125 5257 a 125 5257 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.6) cvn /DEST pdfmark end 125 5257 a 0 TeXcolorgray FL(Corollary)36 b(6.6.)p 0 TeXcolorgray 42 w Fp(If)d FK(gcd\()p FD(n;)15 b(r)s FK(\))25 b(=)g(1)p Fp(,)33 b(then)1496 5413 y FC(j)1548 5390 y FK(~)1521 5413 y FD(N)1594 5427 y Fx(n;r)1695 5413 y FK(\()p FD(k)s(;)15 b(l)r FK(\))p FC(j)27 b FK(=)e FD(n)2087 5375 y Fx(r)r Fs(\000)p FJ(1)2214 5413 y FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 25 25 TeXDict begin HPSdict begin 25 24 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.25) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(25)p 0 TeXcolorgray 0 TeXcolorgray 125 321 a Fp(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(Clearly)d(,)28 b Fz(Z)802 288 y Fx(r)802 343 y(n)870 321 y FK(=)966 235 y Fx(n)p Fs(\000)p FJ(1)995 253 y Ft(F)977 405 y Fx(l)q FJ(=0)1141 298 y FK(~)1114 321 y FD(N)1187 335 y Fx(n;r)1288 321 y FK(\(0)p FD(;)15 b(l)r FK(\).)41 b(Since)28 b(all)f(the)h(elemen)m(ts)h(in)e(the)i(decomp)s(osition)e (ha)m(v)m(e)i(the)125 561 y(same)h(cardinalit)m(y)-8 b(,)30 b(w)m(e)h(get)g(that)g FD(n)1362 528 y Fx(r)1425 561 y FK(=)25 b FC(j)p Fz(Z)1611 528 y Fx(r)1611 583 y(n)1654 561 y FC(j)g FK(=)1800 474 y Fx(n)p Fs(\000)p FJ(1)1819 492 y Ft(P)1811 645 y Fx(l)q FJ(=0)1975 538 y FK(~)1948 561 y FD(N)2021 575 y Fx(n;r)2122 561 y FK(\(0)p FD(;)15 b(l)r FK(\))26 b(=)f FD(n)p FC(j)2535 538 y FK(~)2508 561 y FD(N)2581 575 y Fx(n;r)2682 561 y FK(\(0)p FD(;)15 b(l)r FK(\))p FC(j)p FK(.)625 b Fo(\003)125 812 y SDict begin H.S end 125 812 a 125 812 a SDict begin 13 H.A end 125 812 a 125 812 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.7) cvn /DEST pdfmark end 125 812 a 0 TeXcolorgray FL(Corollary)40 b(6.7.)p 0 TeXcolorgray 44 w Fp(If)35 b FK(gcd\()p FD(n;)15 b(r)s FK(\))31 b(=)g(1)36 b Fp(then,)g(for)g(any) h(p)-5 b(artition)37 b FD(\031)d FK(=)c FC(f)q FK(1)2670 779 y Fx(e)2703 788 y FG(1)2742 812 y FD(;)15 b FK(2)2827 779 y Fx(e)2860 788 y FG(2)2899 812 y FD(;)g(:)g(:)g(:)i(;)e(r)3145 779 y Fx(e)3178 787 y Fq(r)3216 812 y FC(g)36 b Fp(we)g(have)125 920 y(that)558 1109 y FC(j)p FD(N)666 1071 y Fx(\031)656 1131 y(n;r)757 1109 y FK(\()p FD(k)s(;)15 b(l)r FK(\))p FC(j)27 b FK(=)1108 1047 y(1)p 1104 1088 55 4 v 1104 1171 a FD(n)1183 981 y Ft(\022)1466 1054 y FD(n)1250 1162 y(e)1292 1176 y FJ(1)1332 1162 y FD(;)15 b(e)1414 1176 y FJ(2)1454 1162 y FD(;)g(:)g(:)g(:)i(;)e(e)1698 1176 y Fx(r)1736 981 y Ft(\023)1829 1109 y FK(=)2385 1047 y(\()p FD(n)20 b FC(\000)g FK(1\)!)p 1935 1088 1208 4 v 1935 1171 a FD(e)1977 1185 y FJ(1)2016 1171 y FK(!)p FD(e)2083 1185 y FJ(2)2123 1171 y FK(!)15 b FC(\001)g(\001)g(\001)i FD(e)2327 1185 y Fx(r)2365 1171 y FK(!\()p FD(n)k FC(\000)f FD(e)2634 1185 y FJ(1)2693 1171 y FC(\000)g FD(:)15 b(:)g(:)22 b FC(\000)e FD(e)3044 1185 y Fx(r)3082 1171 y FK(\)!)3152 1109 y FD(:)p 0 TeXcolorgray 125 1416 a Fp(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(First)24 b(of)h(all,)f(observ)m(e)h(that)g(the)g (isomorphism)c(of)k(the)f(pro)s(of)g(of)h(Prop)s(osition)3111 1416 y SDict begin H.S end 3111 1416 a 0 TeXcolorgray FK(6.4)p 0 TeXcolorgray 3227 1358 a SDict begin H.R end 3227 1358 a 3227 1416 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6.4) cvn H.B /ANN pdfmark end 3227 1416 a 24 w FK(preserv)m(es)125 1526 y(the)k(strata)569 1503 y(~)543 1526 y FD(N)626 1493 y Fx(\031)616 1548 y(n;r)716 1526 y FK(\()p FD(k)s(;)15 b(l)r FK(\),)31 b(so)f FC(j)1124 1503 y FK(~)1097 1526 y FD(N)1180 1493 y Fx(\031)1170 1548 y(n;r)1271 1526 y FK(\()p FD(k)s(;)15 b(l)r FK(\))p FC(j)31 b FK(is)d(indep)s(enden)m(t)g(of)h FD(k)k FK(and)c FD(l)r FK(.)40 b(It)30 b(also)f(comm)m(utes)i(with)d(the)125 1653 y(action)i(of)h FD(S)557 1667 y Fx(r)625 1653 y FK(so)f FC(j)p FD(N)844 1620 y Fx(\031)834 1676 y(n;r)935 1653 y FK(\()p FD(k)s(;)15 b(l)r FK(\))p FC(j)27 b FK(=)e FC(j)1323 1630 y FK(~)1297 1653 y FD(N)1380 1620 y Fx(\031)1370 1676 y(n;r)1470 1653 y FK(\()p FD(k)s(;)15 b(l)r FK(\))p FD(=S)1760 1667 y Fx(r)1800 1653 y FC(j)30 b FK(is)g(also)g(indep)s (enden)m(t)e(of)j FD(k)i FK(and)d FD(l)r FK(.)224 1797 y(A)m(t)41 b(this)e(p)s(oin)m(t,)j(the)e(pro)s(of)f(is)g(analogous)i (to)f(the)h(one)f(of)g(Corollary)2759 1797 y SDict begin H.S end 2759 1797 a 0 TeXcolorgray FK(6.6)p 0 TeXcolorgray 2875 1738 a SDict begin H.R end 2875 1738 a 2875 1797 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6.6) cvn H.B /ANN pdfmark end 2875 1797 a FK(,)i(but)e(no)m(w)g(observ)m(e) 125 1961 y(that)329 1875 y Fx(n)p Fs(\000)p FJ(1)357 1893 y Ft(F)339 2045 y Fx(l)q FJ(=0)477 1961 y FD(N)560 1928 y Fx(\031)550 1983 y(n;r)650 1961 y FK(\(0)p FD(;)15 b(l)r FK(\))39 b(is)e(the)h(collection)f(of)h(tuples)f(\()p FD(\013)2027 1975 y FJ(1)2067 1961 y FD(;)15 b(:)g(:)g(:)h(;)f(\013) 2326 1975 y Fx(r)2365 1961 y FK(\))38 b FC(2)f Fz(Z)2601 1928 y Fx(r)2601 1983 y(n)2643 1961 y FK(,)j(up)d(to)h(reordering,)h (with)125 2125 y FD(e)167 2139 y FJ(1)245 2125 y FK(collections)g(of)g (di\013eren)m(t)f(elemen)m(ts,)k FD(e)1623 2139 y FJ(2)1702 2125 y FK(collections)c(of)h(2)h(equal)e(elemen)m(ts)i(and,)g(in)e (general,)125 2233 y FD(e)167 2247 y Fx(i)233 2233 y FK(collections)g(of)h FD(i)f FK(equal)g(elemen)m(ts,)j(for)d(1)h FC(\024)g FD(i)f FC(\024)h FD(r)s FK(.)64 b(The)38 b(total)h(coun)m(t)g (of)f(these)h(sets)g(is)e(the)125 2391 y(m)m(ultinomial)27 b(n)m(um)m(b)s(er)963 2263 y Ft(\022)1246 2336 y FD(n)1030 2444 y(e)1072 2458 y FJ(1)1112 2444 y FD(;)15 b(e)1194 2458 y FJ(2)1234 2444 y FD(;)g(:)g(:)g(:)i(;)e(e)1478 2458 y Fx(r)1516 2263 y Ft(\023)1583 2391 y FK(,)31 b(and)f(the)g (result)f(follo)m(ws.)1022 b Fo(\003)224 2666 y FK(With)30 b(this)f(result,)h(w)m(e)h(are)f(ready)h(to)g(pro)m(vide)e(the)i (\014nal)e(coun)m(t.)125 2856 y SDict begin H.S end 125 2856 a 125 2856 a SDict begin 13 H.A end 125 2856 a 125 2856 a SDict begin [/View [/XYZ H.V]/Dest (theorem.6.8) cvn /DEST pdfmark end 125 2856 a 0 TeXcolorgray FL(Theorem)39 b(6.8.)p 0 TeXcolorgray 44 w Fp(If)e FK(gcd\()p FD(n;)15 b(r)s FK(\))33 b(=)f(gcd)q(\()p FD(m;)15 b(r)s FK(\))33 b(=)f(1)p Fp(,)38 b(then,)h(for)e(any)g FD(\031)f FK(=)2733 2782 y Ft(\010)2786 2856 y FK(1)2831 2823 y Fx(e)2864 2832 y FG(1)2903 2856 y FD(;)15 b FK(2)2988 2823 y Fx(e)3021 2832 y FG(2)3061 2856 y FD(;)g(:)g(:)g(:)h(;)f(r)3306 2823 y Fx(e)3339 2831 y Fq(r)3378 2782 y Ft(\011)3467 2856 y Fp(and)125 2980 y FD(\031)180 2947 y Fs(0)228 2980 y FK(=)324 2907 y Ft(\010)377 2980 y FK(1)422 2947 y Fx(e)455 2924 y Fr(0)455 2968 y FG(1)494 2980 y FD(;)g FK(2)579 2947 y Fx(e)612 2924 y Fr(0)612 2968 y FG(2)651 2980 y FD(;)g(:)g(:)g(:)i(;)e(r)897 2947 y Fx(e)930 2924 y Fr(0)930 2964 y Fq(r)969 2907 y Ft(\011)1022 2980 y Fp(,)32 b(we)h(have)914 3239 y FC(j)p FD(M)1037 3201 y Fx(\031)r(;\031)1143 3178 y Fr(0)1027 3261 y Fx(n;m;r)1210 3239 y FC(j)26 b FK(=)1412 3177 y FD(r)p 1367 3218 135 4 v 1367 3301 a(nm)1526 3111 y Ft(\022)1809 3184 y FD(n)1593 3292 y(e)1635 3306 y FJ(1)1675 3292 y FD(;)15 b(e)1757 3306 y FJ(2)1797 3292 y FD(;)g(:)g(:)g(:)h(;)f(e)2040 3306 y Fx(r)2079 3111 y Ft(\023)g(\022)2431 3184 y FD(m)2228 3292 y(e)2270 3259 y Fs(0)2270 3316 y FJ(1)2310 3292 y FD(;)g(e)2392 3259 y Fs(0)2392 3316 y FJ(2)2432 3292 y FD(;)g(:)g(:)g(:)i(;)e(e)2676 3259 y Fs(0)2676 3314 y Fx(r)2714 3111 y Ft(\023)2796 3239 y FD(:)p 0 TeXcolorgray 125 3578 a Fp(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 42 w FK(Observ)m(e)38 b(that)g FC(j)p FD(M)1089 3534 y Fx(\031)r(;\031)1195 3511 y Fr(0)1079 3590 y Fx(n;m;r)1261 3578 y FC(j)f FK(=)1431 3510 y Ft(P)1527 3605 y Fx($)1607 3578 y FC(j)p FD(N)1715 3545 y Fx(\031)1705 3601 y(n;r)1805 3578 y FK(\()p FD($)s FK(\))p FC(j)15 b(j)p FD(N)2101 3545 y Fx(\031)2144 3522 y Fr(0)2091 3601 y Fx(m;r)2213 3578 y FK(\()p FD($)s FK(\))p FC(j)p FK(.)62 b(Hence,)40 b(adding)c(up)g(and)h(using)125 3690 y(that)31 b FC(j)p FD(N)430 3657 y Fx(\031)420 3713 y(n;r)520 3690 y FK(\()p FD($)s FK(\))p FC(j)26 b FK(=)f FC(j)p FD(N)923 3657 y Fx(\031)913 3713 y(n;r)1014 3690 y FK(\()p FD(k)s(;)15 b FK(0\))p FC(j)32 b FK(is)e(indep)s(enden)m(t)e (of)i FD($)s FK(,)h(w)m(e)f(get)564 3916 y FC(j)p FD(M)687 3878 y Fx(\031)r(;\031)793 3855 y Fr(0)677 3938 y Fx(n;m;r)859 3916 y FC(j)c FK(=)1033 3830 y Ft(X)1006 4020 y Fx($)r Fs(2)p Fx(\026)1156 4028 y Fq(r)1206 3916 y FC(j)p FD(N)1314 3878 y Fx(\031)1304 3938 y(n;r)1404 3916 y FK(\()p FD($)s FK(\))p FC(j)21 b(\001)g(j)p FD(N)1752 3878 y Fx(\031)1795 3855 y Fr(0)1742 3938 y Fx(m;r)1862 3916 y FK(\()p FD($)s FK(\))p FC(j)26 b FK(=)f FD(r)s FC(j)p FD(N)2309 3878 y Fx(\031)2299 3938 y(n;r)2399 3916 y FK(\()p FD(k)s(;)15 b FK(0\))p FC(j)22 b(\001)f(j)p FD(N)2805 3878 y Fx(\031)2848 3855 y Fr(0)2795 3938 y Fx(m;r)2915 3916 y FK(\()p FD(k)s(;)15 b FK(0\))p FC(j)p FD(:)125 4200 y FK(No)m(w)31 b(the)f(result)f(follo)m (ws)h(from)g(Corollary)1657 4200 y SDict begin H.S end 1657 4200 a 0 TeXcolorgray FK(6.7)p 0 TeXcolorgray 1773 4142 a SDict begin H.R end 1773 4142 a 1773 4200 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6.7) cvn H.B /ANN pdfmark end 1773 4200 a FK(.)1743 b Fo(\003)224 4426 y FK(All)35 b(form)m(ulas)h(\()790 4426 y SDict begin H.S end 790 4426 a 0 TeXcolorgray FK(35)p 0 TeXcolorgray 881 4367 a SDict begin H.R end 881 4367 a 881 4426 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.35) cvn H.B /ANN pdfmark end 881 4426 a FK(\),)i(\()1014 4426 y SDict begin H.S end 1014 4426 a 0 TeXcolorgray FK(36)p 0 TeXcolorgray 1106 4367 a SDict begin H.R end 1106 4367 a 1106 4426 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.36) cvn H.B /ANN pdfmark end 1106 4426 a FK(\),)g(\()1239 4426 y SDict begin H.S end 1239 4426 a 0 TeXcolorgray FK(37)p 0 TeXcolorgray 1330 4367 a SDict begin H.R end 1330 4367 a 1330 4426 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.37) cvn H.B /ANN pdfmark end 1330 4426 a FK(\),)h(\()1464 4426 y SDict begin H.S end 1464 4426 a 0 TeXcolorgray FK(38)p 0 TeXcolorgray 1555 4367 a SDict begin H.R end 1555 4367 a 1555 4426 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.38) cvn H.B /ANN pdfmark end 1555 4426 a FK(\),)g(\()1689 4426 y SDict begin H.S end 1689 4426 a 0 TeXcolorgray FK(39)p 0 TeXcolorgray 1780 4367 a SDict begin H.R end 1780 4367 a 1780 4426 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.39) cvn H.B /ANN pdfmark end 1780 4426 a FK(\))e(matc)m(h)g(exactly)g(Theorem)2842 4426 y SDict begin H.S end 2842 4426 a 0 TeXcolorgray FK(6.8)p 0 TeXcolorgray 2958 4367 a SDict begin H.R end 2958 4367 a 2958 4426 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6.8) cvn H.B /ANN pdfmark end 2958 4426 a FK(.)59 b(W)-8 b(e)38 b(conjecture)125 4534 y(that)31 b(the)g(form)m(ula)f(of)g (Theorem)1301 4534 y SDict begin H.S end 1301 4534 a 0 TeXcolorgray FK(6.8)p 0 TeXcolorgray 1417 4475 a SDict begin H.R end 1417 4475 a 1417 4534 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6.8) cvn H.B /ANN pdfmark end 1417 4534 a 31 w FK(also)g(holds)g(true)g(for)h(general)f FD(n;)15 b(m)31 b FK(with)e(gcd\()p FD(n;)15 b(m)p FK(\))27 b(=)e(1.)42 b(It)125 4642 y(is)29 b(true)h(for)g FD(r)e FC(\024)d FK(4.)125 4869 y SDict begin H.S end 125 4869 a 125 4869 a SDict begin 13 H.A end 125 4869 a 125 4869 a SDict begin [/View [/XYZ H.V]/Dest (section.7) cvn /DEST pdfmark end 125 4869 a 678 5000 a FK(7.)46 b FE(Irreducible)32 b(chara)n(cter)g(v)-8 b(arieties)33 b(f)n(or)h FK(SL)2635 5014 y FJ(2)2709 5000 y FE(and)g FK(SL)3019 5014 y FJ(3)224 5197 y FK(In)g(this)g(section,)j(w)m(e)e(will)e(use)h(the)i(previous)d (framew)m(ork)i(for)g(computing)f(the)h(motiv)m(e)h(of)f(the)125 5305 y(c)m(haracter)i(v)-5 b(ariet)m(y)36 b(of)g(torus)g(knots)f(in)g (the)h(cases)h(of)f(rank)f(2)h(and)f(rank)g(3.)58 b(These)35 b(results)g(w)m(ere)125 5413 y(\014rst)29 b(obtained)h(in)f(the)i(pap)s (ers)e([)1265 5413 y SDict begin H.S end 1265 5413 a 0 TeXcolorgray FK(27)p 0 TeXcolorgray 1356 5354 a SDict begin H.R end 1356 5354 a 1356 5413 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Munoz) cvn H.B /ANN pdfmark end 1356 5413 a FK(])h(and)g([)1613 5413 y SDict begin H.S end 1613 5413 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 1704 5354 a SDict begin H.R end 1704 5354 a 1704 5413 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 1704 5413 a FK(],)h(resp)s(ectiv)m(ely)-8 b(.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 26 26 TeXDict begin HPSdict begin 26 25 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.26) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(26)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 125 183 a SDict begin H.S end 125 183 a 125 183 a SDict begin 13 H.A end 125 183 a 125 183 a SDict begin [/View [/XYZ H.V]/Dest (subsection.7.1) cvn /DEST pdfmark end 125 183 a 83 x FK(7.1.)46 b FL(Rank)g FK(2)p FL(.)g FK(By)40 b(Remark)1222 266 y SDict begin H.S end 1222 266 a 0 TeXcolorgray FK(3.5)p 0 TeXcolorgray 1338 207 a SDict begin H.R end 1338 207 a 1338 266 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.5) cvn H.B /ANN pdfmark end 1338 266 a FK(,)i(the)e(only)e (con\014guration)h(of)h(eigen)m(v)-5 b(alues)39 b(that)h(admits)e (irre-)125 374 y(ducible)31 b(represen)m(tations)j(is)f FD(\024)f FK(=)f(\()p FC(f)q FD(\017)1474 388 y FJ(1)1513 374 y FD(;)15 b(\017)1590 388 y FJ(2)1630 374 y FC(g)h FD(;)f FC(f)q FD(")1819 388 y FJ(1)1858 374 y FD(;)g(")1940 388 y FJ(2)1981 374 y FC(g)p FK(\))34 b(with)f FD(\017)2343 388 y FJ(1)2414 374 y FC(6)p FK(=)e FD(\017)2553 388 y FJ(2)2626 374 y FK(and)j FD(")2849 388 y FJ(1)2920 374 y FC(6)p FK(=)d FD(")3064 388 y FJ(2)3104 374 y FK(.)51 b(T)-8 b(o)35 b(shorten)125 511 y(the)c(notation,)h(for)f(a)h(t)m(yp)s (e)f FD(\034)37 b FK(=)26 b(\()p FD(\030)t(;)15 b(\033)s FK(\))32 b(with)e(shap)s(e)h FD(\030)f FK(=)2142 411 y Ft(\020)2196 511 y FC(f)q FK(\()p FD(d)2324 525 y Fx(i;j)2404 511 y FD(;)15 b(m)2524 525 y Fx(i;j)2605 511 y FK(\))p FC(g)2686 470 y Fx(s)2719 480 y Fq(i)2686 542 y Fx(j)t FJ(=1)2812 411 y Ft(\021)2867 433 y Fx(s)2867 575 y(i)p FJ(=1)2985 511 y FK(,)32 b(w)m(e)f(will)e(denote)125 646 y(b)m(y)40 b FD(\033)313 660 y Fx(A)410 646 y FK(and)f FD(\033)648 660 y Fx(B)749 646 y FK(the)h(con\014guration)g(of)g(eigen) m(v)-5 b(alues)40 b(of)h(eac)m(h)g(irreducible)c(piece)j(for)g FD(A)g FK(and)g FD(B)125 754 y FK(resp)s(ectiv)m(ely)-8 b(.)224 897 y(In)30 b(this)f(case,)j FC(T)783 911 y Fx(\024)827 897 y FD(=S)928 911 y Fx(\024)1003 897 y FK(con)m(tains)f(the)g(follo)m (wing)d(t)m(w)m(o)k(t)m(yp)s(es)e(of)h(the)g(form)e FD(\034)36 b FK(=)25 b(\()p FD(\030)t(;)15 b(\033)s FK(\).)p 0 TeXcolorgray 340 1116 a FC(\017)p 0 TeXcolorgray 42 w FD(\030)43 b FK(=)38 b(\()p FC(f)q FK(\(1)p FD(;)15 b FK(1\))p FD(;)g FK(\(1)p FD(;)g FK(1\))p FC(g)5 b FK(\))38 b(with)f(eigen)m(v)-5 b(alues)38 b(p)s(er)f(piece)i FD(\033)2411 1130 y Fx(A)2506 1116 y FK(=)f(\()p FC(f)q(f)p FD(\017)2778 1130 y FJ(1)2818 1116 y FC(g)15 b FD(;)g FC(f)q FD(\017)3001 1130 y FJ(2)3041 1116 y FC(gg)p FK(\))39 b(and)f FD(\033)3442 1130 y Fx(B)3541 1116 y FK(=)427 1223 y(\()p FC(f)q(f)q FD(")596 1237 y FJ(1)635 1223 y FC(g)16 b FD(;)f FC(f)q FD(")824 1237 y FJ(2)864 1223 y FC(gg)q FK(\).)41 b(F)-8 b(or)31 b(this)e(t)m(yp)s (e,)i(w)m(e)g(ha)m(v)m(e)782 1486 y FD(m)862 1500 y Fx(\024)907 1486 y FK(\()p FD(\034)10 b FK(\))26 b(=)f(2)p FD(;)107 b FK([)p FB(M)1446 1449 y FJ(irr)1446 1509 y Fx(\034)1525 1486 y FK(])26 b(=)e(1)p FD(;)107 b FK([)p FC(M)1982 1500 y Fx(\034)2026 1486 y FK(])25 b(=)g(1)p FD(;)107 b FK([)p FC(G)2428 1500 y Fx(\034)2471 1486 y FK(])26 b(=)f(\()p FD(q)e FC(\000)d FK(1\))2888 1449 y FJ(2)2929 1486 y FD(:)427 1767 y FK(Therefore)30 b([)p FD(R)q FK(\()p FD(\034)10 b FK(\)])27 b(=)e(2)1257 1722 y Fx(q)r FJ(\()p Fx(q)r FJ(+1\)\()p Fx(q)r Fs(\000)p FJ(1\))1647 1699 y FG(2)p 1257 1746 427 4 v 1363 1801 a FJ(\()p Fx(q)r Fs(\000)p FJ(1\))1541 1782 y FG(2)1719 1767 y FK(=)g(2)p FD(q)1904 1734 y FJ(2)1964 1767 y FK(+)19 b(2)p FD(q)s FK(.)p 0 TeXcolorgray 340 1894 a FC(\017)p 0 TeXcolorgray 42 w FD(\030)51 b FK(=)46 b(\()p FC(f)q FK(\(1)p FD(;)15 b FK(1\))p FC(g)j FD(;)d FC(f)q FK(\(1)p FD(;)g FK(1\))p FC(g)r FK(\))44 b(with)e(eigen)m(v)-5 b(alues)43 b(p)s(er)f(piece)h FD(\033)2558 1908 y Fx(A)2662 1894 y FK(=)j(\()p FC(f)q(f)p FD(\017)2942 1908 y FJ(1)2982 1894 y FC(gg)16 b FD(;)f FC(f)q(f)p FD(\017)3256 1908 y FJ(2)3296 1894 y FC(gg)p FK(\))44 b(and)427 2002 y FD(\033)479 2016 y Fx(B)565 2002 y FK(=)25 b(\()p FC(f)q(f)p FD(")829 2016 y FJ(1)869 2002 y FC(g)q(g)15 b FD(;)g FC(f)q(f)q FD(")1149 2016 y FJ(2)1188 2002 y FC(g)q(g)p FK(\).)41 b(In)30 b(this)f(case,)j(w)m(e) f(ha)m(v)m(e)705 2264 y FD(m)785 2278 y Fx(\024)829 2264 y FK(\()p FD(\034)10 b FK(\))26 b(=)f(4)p FD(;)107 b FK([)p FB(M)1368 2227 y FJ(irr)1368 2287 y Fx(\034)1448 2264 y FK(])25 b(=)g(1)p FD(;)107 b FK([)p FC(M)1905 2278 y Fx(\034)1948 2264 y FK(])26 b(=)f FD(q)e FC(\000)d FK(1)p FD(;)107 b FK([)p FC(G)2506 2278 y Fx(\034)2549 2264 y FK(])25 b(=)g(\()p FD(q)f FC(\000)c FK(1\))2966 2227 y FJ(2)3006 2264 y FD(:)427 2545 y FK(Therefore)30 b([)p FD(R)q FK(\()p FD(\034)10 b FK(\)])27 b(=)e(4\()p FD(q)e FC(\000)d FK(1\))1527 2500 y Fx(q)r FJ(\()p Fx(q)r FJ(+1\)\()p Fx(q)r Fs(\000)p FJ(1\))1917 2477 y FG(2)p 1528 2524 V 1635 2579 a FJ(\()p Fx(q)r Fs(\000)p FJ(1\))1813 2560 y FG(2)1990 2545 y FK(=)25 b(4)p FD(q)2175 2512 y FJ(3)2235 2545 y FC(\000)20 b FK(4)p FD(q)s FK(.)224 2793 y(Putting)39 b(this)f(all)h(together,)k(w)m(e)e(get)f(that)g([)p FD(R)1890 2760 y FJ(red)1889 2816 y Fx(\024)1993 2793 y FK(])g(=)h(4)p FD(q)2259 2760 y FJ(3)2325 2793 y FK(+)26 b(2)p FD(q)2511 2760 y FJ(2)2577 2793 y FC(\000)g FK(2)p FD(q)s FK(.)69 b(The)39 b(total)h(coun)m(t)g(of)125 2945 y(represen)m(tations)g(is)f([)p FD(R)960 2959 y Fx(\024)1006 2945 y FK(])j(=)1186 2844 y Ft(\020)1250 2900 y Fx(q)r FJ(\()p Fx(q)r Fs(\000)p FJ(1\)\()p Fx(q)r Fs(\000)p FJ(1\))1640 2877 y FG(2)p 1250 2924 V 1356 2979 a FJ(\()p Fx(q)r Fs(\000)p FJ(1\))1534 2960 y FG(2)1686 2844 y Ft(\021)1741 2867 y FJ(2)1822 2945 y FK(=)g FD(q)1979 2912 y FJ(4)2045 2945 y FK(+)27 b(2)p FD(q)2232 2912 y FJ(3)2298 2945 y FK(+)g FD(q)2440 2912 y FJ(2)2479 2945 y FK(.)71 b(Therefore,)43 b(the)e(irreducible)125 3092 y(represen)m(tations)35 b(are)h([)p FD(R)1012 3059 y FJ(irr)1011 3114 y Fx(\024)1091 3092 y FK(])e(=)g([)p FD(R)1349 3106 y Fx(\024)1394 3092 y FK(])24 b FC(\000)f FK([)p FD(R)1632 3059 y FJ(red)1631 3114 y Fx(\024)1734 3092 y FK(])34 b(=)g FD(q)1942 3059 y FJ(4)2005 3092 y FC(\000)23 b FK(2)p FD(q)2188 3059 y FJ(3)2252 3092 y FC(\000)g FD(q)2390 3059 y FJ(2)2453 3092 y FK(+)g(2)p FD(q)s FK(.)57 b(Moreo)m(v)m(er,)39 b(w)m(e)d(get)h(that)125 3200 y([)p FB(M)245 3167 y FJ(irr)245 3223 y Fx(\024)324 3200 y FK(])25 b(=)g([)p FD(R)565 3167 y FJ(irr)564 3223 y Fx(\024)644 3200 y FK(])p FD(=)p FK([PGL)930 3214 y FJ(2)970 3200 y FK(])g(=)g FD(q)e FC(\000)d FK(2.)224 3344 y(Finally)-8 b(,)32 b(b)m(y)h(\()718 3344 y SDict begin H.S end 718 3344 a 0 TeXcolorgray FK(35)p 0 TeXcolorgray 810 3285 a SDict begin H.R end 810 3285 a 810 3344 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.6.35) cvn H.B /ANN pdfmark end 810 3344 a FK(\))g(w)m(e)g(kno)m(w)h(ho)m(w)f (man)m(y)g(con\014gurations)f(of)h(eigen)m(v)-5 b(alues)33 b FD(\024)g FK(exist.)49 b(This)31 b(giv)m(es)125 3452 y(the)f(n)m(um)m(b)s(er)f(of)i(comp)s(onen)m(ts)f(of)h FB(M)1415 3419 y FJ(irr)1415 3477 y(2)1524 3452 y FK(of)g(the)f(form)g FB(M)2094 3419 y FJ(irr)2094 3475 y Fx(\024)2173 3452 y FK(,)h(so)g(follo)m(wing)d(\()2759 3452 y SDict begin H.S end 2759 3452 a 0 TeXcolorgray FK(12)p 0 TeXcolorgray 2851 3394 a SDict begin H.R end 2851 3394 a 2851 3452 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.3.12) cvn H.B /ANN pdfmark end 2851 3452 a FK(\))i(w)m(e)h(get)h(that)633 3766 y([)p FB(M)753 3729 y FJ(irr)753 3789 y(2)833 3766 y FK(])25 b(=)989 3705 y(1)p 989 3745 46 4 v 989 3828 a(2)1045 3638 y Ft(\022)1111 3705 y FD(n)20 b FC(\000)g FK(1)1116 3828 y(2)h FC(\000)f FK(1)1323 3638 y Ft(\023)o(\022)1456 3705 y FD(m)g FC(\000)g FK(1)1474 3828 y(2)g FC(\000)g FK(1)1693 3638 y Ft(\023)1760 3766 y FK(\()p FD(q)j FC(\000)d FK(2\))26 b(=)2162 3705 y(1)p 2162 3745 V 2162 3828 a(2)2217 3766 y(\()p FD(n)21 b FC(\000)e FK(1\)\()p FD(m)j FC(\000)e FK(1\)\()p FD(q)k FC(\000)c FK(2\))p FD(:)125 4130 y SDict begin H.S end 125 4130 a 125 4130 a SDict begin 13 H.A end 125 4130 a 125 4130 a SDict begin [/View [/XYZ H.V]/Dest (subsection.7.2) cvn /DEST pdfmark end 125 4130 a 107 x FK(7.2.)46 b FL(Rank)c FK(3)p FL(.)k FK(In)36 b(this)f(case,)k(there)e(are)f(three)h(di\013eren)m(t)e(t)m(yp)s(es)i (of)f(p)s(ossible)e(con\014gurations)h(of)125 4345 y(eigen)m(v)-5 b(alues,)49 b(according)c(to)h(their)f(m)m(ultiplicit)m(y)-8 b(.)83 b(Observ)m(e)46 b(that)g(there)f(cannot)h(b)s(e)f(a)h(double)125 4453 y(eigen)m(v)-5 b(alue)31 b(b)s(oth)f(in)g(the)i(matrices)f FD(A)g FK(and)g FD(B)36 b FK(since,)31 b(in)f(that)i(case,)g(the)g (represen)m(tation)f(is)f(auto-)125 4561 y(matically)25 b(reducible)e(b)m(y)j(Remark)1362 4561 y SDict begin H.S end 1362 4561 a 0 TeXcolorgray FK(3.5)p 0 TeXcolorgray 1478 4502 a SDict begin H.R end 1478 4502 a 1478 4561 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.5) cvn H.B /ANN pdfmark end 1478 4561 a FK(.)39 b(In)25 b(the)h(follo)m(wing,)g FD(\017)2246 4575 y FJ(1)2285 4561 y FD(;)15 b(\017)2362 4575 y FJ(2)2402 4561 y FD(;)g(\017)2479 4575 y FJ(3)2545 4561 y FK(\(resp.)26 b FD(")2836 4575 y FJ(1)2876 4561 y FD(;)15 b(")2958 4575 y FJ(2)2998 4561 y FD(;)g(")3080 4575 y FJ(3)3120 4561 y FK(\))26 b(will)e(denote)125 4668 y(three)30 b(di\013eren)m(t)g(eigen)m(v)-5 b(alues.)125 4964 y SDict begin H.S end 125 4964 a 125 4964 a SDict begin 13 H.A end 125 4964 a 125 4964 a SDict begin [/View [/XYZ H.V]/Dest (subsubsection.7.2.1) cvn /DEST pdfmark end 125 4964 a 125 x FK(7.2.1.)47 b FD(\024)434 5103 y FJ(1)499 5089 y FK(=)25 b(\(\()p FD(\017)702 5103 y FJ(1)742 5089 y FD(;)15 b(\017)819 5103 y FJ(2)859 5089 y FD(;)g(\017)936 5103 y FJ(3)976 5089 y FK(\))p FD(;)g FK(\()p FD(")1128 5103 y FJ(1)1169 5089 y FD(;)g(")1251 5103 y FJ(2)1291 5089 y FD(;)g(")1373 5103 y FJ(3)1413 5089 y FK(\)\))p Fp(.)46 b FK(In)25 b(this)f(case,)j(the)f(three)f(eigen)m(v)-5 b(alues)25 b(are)h(di\013eren)m(t.)38 b(This)125 5197 y(implies)24 b(that)j FC(G)677 5211 y Fx(\034)747 5197 y FK(is)g(a)g(torus)g(for)f(an)m(y)i(t)m(yp)s(e)f FD(\034)35 b FC(2)25 b(T)1854 5211 y Fx(\024)1895 5220 y FG(1)1960 5197 y FK(of)i(rank)f(equal)h(to)h(the)f(n)m(um)m(b)s(er)f(of)h (irreducible)125 5305 y(pieces.)38 b(In)23 b(suc)m(h)h(manner,)h(the)g (con)m(tributions)d(of)j(the)f(p)s(ossible)e(t)m(yp)s(es)i(are)h(sho)m (wn)f(in)e(the)j(follo)m(wing)125 5413 y(table.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 27 27 TeXDict begin HPSdict begin 27 26 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 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1767 y Fi(g)13 b Fj(;)h Fi(f)p Fj(")1240 1779 y Ff(2)1277 1767 y Fi(g)o(g)f Fj(;)h Fi(ff)o Fj(")1532 1779 y Ff(3)1569 1767 y Fi(gg)o Fe(\))p 1814 1797 V 1909 1668 a Fj(q)1949 1638 y Ff(2)2005 1668 y Fi(\000)k Fe(1)p 2221 1797 V 155 w(\()p Fj(q)j Fi(\000)d Fe(1\))2532 1638 y Ff(3)p 2629 1797 V 2743 1668 a Fe(1)p 2895 1797 V 246 w(3)p 3205 1797 V 530 1800 2677 4 v 528 2099 4 299 v 661 1870 a Fj(\030)27 b Fe(=)c(\()p Fi(f)p Fe(\(1)p Fj(;)14 b Fe(1\))p Fj(;)g Fe(\(1)p Fj(;)g Fe(1\))p Fi(g)e Fj(;)i Fi(f)o Fe(\(1)p Fj(;)g Fe(1\))p Fi(g)o Fe(\))669 1969 y Fj(\033)716 1981 y Fh(A)794 1969 y Fe(=)23 b(\()p Fi(f)o(f)p Fj(\017)1031 1981 y Ff(1)1068 1969 y Fi(g)13 b Fj(;)h Fi(f)p Fj(\017)1236 1981 y Ff(2)1273 1969 y Fi(g)o(g)f Fj(;)h Fi(ff)o Fj(\017)1523 1981 y Ff(1)1560 1969 y Fi(gg)o Fe(\))660 2069 y Fj(\033)707 2081 y Fh(B)788 2069 y Fe(=)23 b(\()p Fi(f)o(f)p Fj(")1030 2081 y Ff(1)1067 2069 y Fi(g)13 b Fj(;)h Fi(f)p Fj(")1240 2081 y Ff(2)1277 2069 y Fi(g)o(g)f Fj(;)h Fi(ff)o Fj(")1532 2081 y 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/DEST pdfmark end 125 1679 a 902 1785 a FK(8.)46 b FE(Irreducible)32 b(chara)n(cter)g(v)-8 b(ariety)33 b(f)n(or)h FK(SL)2795 1799 y FJ(4)224 1983 y FK(In)i(this)g(section,)j(w)m(e)f(compute)f(the)g(motiv)m(e)h(of)f (the)g(irreducible)d(SL)2686 1997 y FJ(4)2725 1983 y FK(-c)m(haracter)39 b(v)-5 b(ariet)m(y)d(.)61 b(As)125 2090 y(in)32 b(Section)556 2090 y SDict begin H.S end 556 2090 a 0 TeXcolorgray FK(7)p 0 TeXcolorgray 602 2032 a SDict begin H.R end 602 2032 a 602 2090 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.7) cvn H.B /ANN pdfmark end 602 2090 a FK(,)j(with)e(the)h(metho)s(ds)f(dev)m(elop)s(ed)h(in)f (this)g(pap)s(er)g(the)h(computation)g(reduces)g(to)g(an)125 2198 y(en)m(umerativ)m(e)d(problem.)39 b(First,)30 b(w)m(e)h(need)f(to) i(consider)d(the)i(p)s(ossible)d(eigen)m(v)-5 b(alues)30 b(con\014gurations)125 2306 y(that)40 b(ma)m(y)g(con)m(tain)f (irreducible)d(represen)m(tations,)42 b(in)d(the)g(sense)h(that)g(they) f(do)h(not)f(ful\014ll)d(the)125 2414 y(conditions)28 b(of)j(Remark)1005 2414 y SDict begin H.S end 1005 2414 a 0 TeXcolorgray FK(3.5)p 0 TeXcolorgray 1122 2356 a SDict begin H.R end 1122 2356 a 1122 2414 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3.5) cvn H.B /ANN pdfmark end 1122 2414 a FK(.)40 b(As)31 b(w)m(e)f(will)e(see,)j(there)g(are) g(11)g(p)s(ossible)d(con\014gurations.)224 2558 y(Among)21 b(these,)i(there)d(is)g(a)g(sp)s(ecial)f(con\014guration,)j(namely)e FD(\024)25 b FK(=)g(\(\()p FD(\017)2589 2572 y FJ(1)2629 2558 y FD(;)15 b(\017)2706 2572 y FJ(1)2746 2558 y FD(;)g(\017)2823 2572 y FJ(2)2863 2558 y FD(;)g(\017)2940 2572 y FJ(2)2979 2558 y FK(\))p FD(;)g FK(\()p FD(")3131 2572 y FJ(1)3172 2558 y FD(;)g(")3254 2572 y FJ(1)3295 2558 y FD(;)g(")3377 2572 y FJ(2)3417 2558 y FD(;)g(")3499 2572 y FJ(2)3539 2558 y FK(\)\).)125 2666 y(This)22 b(con\014guration)i(is)f(the)h(only) g(one)g(that)h(con)m(tains)f(t)m(yp)s(es)h FD(\034)35 b FC(2)25 b(T)2447 2680 y Fx(\024)2515 2666 y FK(with)e(isot)m(ypic)h (comp)s(onen)m(ts)g(of)125 2773 y(dimension)19 b(greater)k(than)e(1)h (and)g(with)e(m)m(ultiplicit)m(y)f(greater)k(than)e(1,)k(namely)-8 b(,)23 b(the)f(t)m(yp)s(e)g FD(\034)35 b FK(=)25 b(\()p FD(\030)t(;)15 b(\033)s FK(\))125 2881 y(with)42 b(shap)s(e)g FD(\030)51 b FK(=)46 b(\()p FC(f)q FK(\(2)p FD(;)15 b FK(2\))p FC(g)s FK(\))43 b(and)g(eigen)m(v)-5 b(alues)43 b FD(\033)1953 2895 y Fx(A)2057 2881 y FK(=)j(\()p FC(f)q(f)p FD(\017)2337 2895 y FJ(1)2377 2881 y FD(;)15 b(\017)2454 2895 y FJ(2)2494 2881 y FC(gg)p FK(\))44 b(and)f FD(\033)2905 2895 y Fx(B)3012 2881 y FK(=)k(\()p FC(ff)q FD(")3298 2895 y FJ(1)3338 2881 y FD(;)15 b(")3420 2895 y FJ(2)3460 2881 y FC(gg)q FK(\).)125 2989 y(Hence,)30 b(this)f(is)f(the)i(only)f (con\014guration)g(for)g(whic)m(h)g(form)m(ula)f(\()2356 2989 y SDict begin H.S end 2356 2989 a 0 TeXcolorgray FK(22)p 0 TeXcolorgray 2448 2931 a SDict begin H.R end 2448 2931 a 2448 2989 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.22) cvn H.B /ANN pdfmark end 2448 2989 a FK(\))i(is)e(not)i(v)-5 b(alid)28 b(b)s(ecause)i(there)f(is)125 3098 y(an)34 b(action)i(of)f Fz(Z)706 3112 y FJ(2)776 3098 y FK(on)f FB(M)1001 3065 y FJ(irr)1001 3120 y Fx(\034)1081 3098 y FK(.)54 b(F)-8 b(ortunately)g(,)37 b(w)m(e)f(do)e(not)i(need)e(to)i (coun)m(t)g(this)d(stratum)i(thanks)g(to)125 3206 y(the)30 b(follo)m(wing)f(result.)125 3372 y SDict begin H.S end 125 3372 a 125 3372 a SDict begin 13 H.A end 125 3372 a 125 3372 a SDict begin [/View [/XYZ H.V]/Dest (theorem.8.1) cvn /DEST pdfmark end 125 3372 a 0 TeXcolorgray FL(Prop)s(osition)37 b(8.1.)p 0 TeXcolorgray 43 w Fp(L)-5 b(et)33 b FD(\024)28 b FK(=)e(\(\()p FD(\017)1337 3386 y FJ(1)1377 3372 y FD(;)15 b(\017)1454 3386 y FJ(1)1494 3372 y FD(;)g(\017)1571 3386 y FJ(2)1611 3372 y FD(;)g(\017)1688 3386 y FJ(2)1728 3372 y FK(\))p FD(;)g FK(\()p FD(")1880 3386 y FJ(1)1921 3372 y FD(;)g(")2003 3386 y FJ(1)2043 3372 y FD(;)g(")2125 3386 y FJ(2)2165 3372 y FD(;)g(")2247 3386 y FJ(2)2287 3372 y FK(\)\))p Fp(.)46 b(Then)33 b FD(R)2738 3339 y FJ(irr)2737 3394 y Fx(\024)2844 3372 y FK(=)27 b FC(;)p Fp(,)34 b(that)g(is,)g(al)5 b(l)34 b(the)125 3480 y(r)-5 b(epr)g(esentations)35 b(of)e FD(R)930 3494 y Fx(\024)1007 3480 y Fp(ar)-5 b(e)34 b(r)-5 b(e)g(ducible.)p 0 TeXcolorgray 125 3682 a(Pr)g(o)g(of.)p 0 TeXcolorgray 42 w FK(Let)43 b FD(\032)j FK(=)f(\()p FD(A;)15 b(B)5 b FK(\))46 b FC(2)f FD(R)1262 3696 y Fx(\024)1349 3682 y FK(and)d(let)g FD(V)1734 3696 y Fx(i)1762 3682 y FD(;)15 b(W)1888 3696 y Fx(i)1962 3682 y FC(\022)45 b FD(k)2128 3649 y FJ(4)2210 3682 y FK(b)s(e)d(the)h(eigenspaces)f(of)h FD(A)g FK(and)e FD(B)47 b FK(of)125 3790 y(eigen)m(v)-5 b(alues)42 b FD(\017)644 3804 y Fx(i)714 3790 y FK(and)g FD(")945 3804 y Fx(i)1016 3790 y FK(resp)s(ectiv)m(ely)-8 b(,)45 b(for)d FD(i)j FK(=)g(1)p FD(;)15 b FK(2.)78 b(W)-8 b(e)44 b(claim)d(that)i(there)f (exists)g FD(v)3303 3804 y FJ(1)3388 3790 y FC(2)j FD(V)3547 3804 y FJ(1)3586 3790 y FK(,)125 3898 y FD(v)169 3912 y FJ(2)233 3898 y FC(2)25 b FD(V)372 3912 y FJ(2)439 3898 y FK(and)j FD(\026)d FC(2)g FD(k)31 b FK(suc)m(h)c(that)i FD(v)1299 3912 y FJ(1)1354 3898 y FK(+)15 b FD(v)1484 3912 y FJ(2)1549 3898 y FC(2)24 b FD(W)1720 3912 y FJ(1)1788 3898 y FK(and)j FD(v)2006 3912 y FJ(1)2061 3898 y FK(+)15 b FD(\026v)2246 3912 y FJ(2)2310 3898 y FC(2)25 b FD(W)2482 3912 y FJ(2)2522 3898 y FK(.)39 b(In)28 b(that)g(case,)i(the)e (subspace)125 4006 y FD(H)42 b FK(=)36 b FC(h)p FD(v)429 4020 y FJ(1)469 4006 y FD(;)15 b(v)553 4020 y FJ(2)593 4006 y FC(i)36 b FK(=)f FC(h)p FD(v)849 4020 y FJ(1)913 4006 y FK(+)25 b FD(v)1053 4020 y FJ(2)1092 4006 y FD(;)15 b(v)1176 4020 y FJ(1)1240 4006 y FK(+)25 b FD(\026v)1435 4020 y FJ(2)1474 4006 y FC(i)37 b FK(is)f(an)g(in)m(v)-5 b(arian)m(t)36 b(subrepresen)m(tation,)i(pro)m(ving)d(that)j FD(\032)e FK(is)125 4114 y(reducible.)i(Observ)m(e)30 b(that)h(w)m(e)g(m)m(ust)f(ha)m(v)m(e)i FD(\026)25 b FC(6)p FK(=)g(1)31 b(since)e(otherwise)h FD(v)2587 4128 y FJ(1)2647 4114 y FK(+)20 b FD(v)2782 4128 y FJ(2)2847 4114 y FC(2)k FD(W)3018 4128 y FJ(1)3078 4114 y FC(\\)c FD(W)3245 4128 y FJ(2)3284 4114 y FK(.)224 4257 y(In)42 b(order)g(to)h(obtain)f(these)h(v)m(ectors,)k(let)42 b FD(L)1817 4271 y Fx(i)1891 4257 y FK(=)j FD(L)2069 4271 y Fx(i;)p FJ(1)2180 4257 y FK(+)28 b FD(L)2341 4271 y Fx(i;)p FJ(2)2469 4257 y FK(:)46 b FD(V)2593 4271 y FJ(1)2660 4257 y FC(\010)28 b FD(V)2812 4271 y FJ(2)2897 4257 y FC(!)45 b FD(k)3083 4224 y FJ(2)3165 4257 y FK(b)s(e)d(a)h (linear)125 4365 y(map)38 b(with)f FD(W)636 4379 y Fx(i)703 4365 y FK(=)h(k)m(er)16 b FD(L)1011 4379 y Fx(i)1077 4365 y FK(for)38 b FD(i)h FK(=)g(1)p FD(;)15 b FK(2.)66 b(Observ)m(e)38 b(that)h(w)m(e)g(can)g(supp)s(ose)e(that)i(the)f(maps)g FD(L)3531 4379 y Fx(i;j)125 4473 y FK(are)33 b(in)m(v)m(ertible)f (since)g(otherwise)h FD(W)1403 4487 y Fx(i)1453 4473 y FC(\\)21 b FD(V)1588 4487 y Fx(j)1655 4473 y FC(6)p FK(=)29 b(0.)50 b(The)32 b(v)m(ectors)j(w)m(e)f(are)f(lo)s(oking)f(for) h(m)m(ust)g(satisfy)125 4581 y FD(L)187 4595 y FJ(1)p Fx(;)p FJ(1)281 4581 y FK(\()p FD(v)360 4595 y FJ(1)400 4581 y FK(\))20 b(+)g FD(L)608 4595 y FJ(1)p Fx(;)p FJ(2)702 4581 y FK(\()p FD(v)781 4595 y FJ(2)821 4581 y FK(\))26 b(=)f(0)30 b(and)g FD(L)1292 4595 y FJ(2)p Fx(;)p FJ(1)1386 4581 y FK(\()p FD(v)1465 4595 y FJ(1)1505 4581 y FK(\))21 b(+)f FD(\026L)1769 4595 y FJ(2)p Fx(;)p FJ(2)1863 4581 y FK(\()p FD(v)1942 4595 y FJ(2)1982 4581 y FK(\))25 b(=)g(0.)41 b(F)-8 b(orm)31 b(the)g(matrix)1464 4792 y FD(L)25 b FK(=)1647 4664 y Ft(\022)1714 4737 y FD(L)1776 4751 y FJ(1)p Fx(;)p FJ(1)1981 4737 y FD(L)2043 4751 y FJ(1)p Fx(;)p FJ(2)1714 4845 y FD(L)1776 4859 y FJ(2)p Fx(;)p FJ(1)1953 4845 y FD(\026L)2070 4859 y FJ(2)p Fx(;)p FJ(2)2164 4664 y Ft(\023)2246 4792 y FD(:)125 4995 y FK(The)31 b(equation)g(det)16 b FD(L)27 b FK(=)g(0)32 b(giv)m(es)g(a)g(quadratic)f(equation)h(for)f FD(\026)p FK(.)45 b(Cho)s(osing)30 b(one)i(of)g(its)f(solutions,)125 5103 y(an)m(y)f FD(v)340 5117 y FJ(1)400 5103 y FC(\010)20 b FD(v)535 5117 y FJ(2)600 5103 y FC(2)k FK(k)m(er)16 b FD(L)30 b FK(pro)m(vides)g(the)g(desired)f(v)m(ectors.)1500 b Fo(\003)224 5305 y FK(With)27 b(this)f(result)h(at)h(hand,)f(it)g(is) f(enough)h(to)i(consider)d(the)i(remaining)d(10)j(p)s(ossible)d (con\014gura-)125 5413 y(tions)e(of)h(eigen)m(v)-5 b(alues.)38 b(As)24 b(w)m(e)g(will)d(see,)26 b(dep)s(ending)21 b(on)j(whether)f (they)h(admit)f(t)m(yp)s(es)g(with)g(isot)m(ypic)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 31 31 TeXDict begin HPSdict begin 31 30 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.31) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(31)p 0 TeXcolorgray 125 266 a FK(comp)s(onen)m(ts)26 b(of)h(higher)e(m)m(ultiplicit)m(y)f(or)i (not,)i(the)f(coun)m(ting)f(metho)s(d)g(is)f(sligh)m(tly)g(di\013eren)m (t,)i(so)g(w)m(e)125 374 y(will)g(analyze)k(them)f(in)f(separate)j (sections.)224 517 y(Observ)m(e)37 b(that,)i(as)e(suggested)g(for)g (the)g(computations)g(of)f(the)h(cases)h(of)f(rank)f(2)h(and)g(rank)f (3,)125 625 y(the)g(n)m(um)m(b)s(er)e(of)i(t)m(yp)s(es)g(to)h(b)s(e)e (analyzed)h(gro)m(ws)g(exp)s(onen)m(tially)e(with)h(the)h(rank.)56 b(In)36 b(the)g(rank)f(2)125 733 y(case,)30 b(w)m(e)e(needed)g(to)i (consider)d(2)i(classes)f(of)g(t)m(yp)s(es,)h(whereas)g(in)e(the)h (rank)g(3)h(case)g(w)m(e)g(had)f(to)h(deal)125 841 y(with)i(23)j (di\013eren)m(t)e(classes)h(of)f(t)m(yp)s(es.)48 b(In)32 b(the)h(rank)g(4)g(case,)h(there)f(are)g(more)g(than)g(350)h(t)m(yp)s (es)f(to)125 949 y(b)s(e)c(analyzed,)h(so)g(the)g(computation)g(m)m (ust)f(b)s(e)h(p)s(erformed)e(with)g(the)i(aid)f(of)h(a)h(computer)e (algebra)125 1057 y(system.)38 b(In)23 b(our)g(case,)j(w)m(e)e(use)g (SageMath)h(and)e(the)g(complete)i(script)d(p)s(erforming)f(the)j (calculating)125 1165 y(can)35 b(b)s(e)f(found)f(in)h([)819 1165 y SDict begin H.S end 819 1165 a 0 TeXcolorgray FK(11)p 0 TeXcolorgray 910 1106 a SDict begin H.R end 910 1106 a 910 1165 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.GPMScript) cvn H.B /ANN pdfmark end 910 1165 a FK(].)54 b(F)-8 b(or)36 b(this)d(reason,)k(in)c(order)h(to)i(k)m(eep)g(this)d (pap)s(er)h(at)h(a)h(reasonable)e(size,)125 1272 y(w)m(e)g(will)d(only) j(rep)s(ort)f(here)h(some)g(particular)f(cases)i(in)d(order)i(to)h (illustrate)d(the)i(metho)s(d.)51 b(F)-8 b(or)35 b(a)125 1380 y(complete)29 b(description)f(of)h(the)h(p)s(ossible)c(t)m(yp)s (es)k(and)e(their)h(coun)m(t,)h(please)f(refer)g(to)h(the)g(script)e([) 3495 1380 y SDict begin H.S end 3495 1380 a 0 TeXcolorgray FK(11)p 0 TeXcolorgray 3586 1322 a SDict begin H.R end 3586 1322 a 3586 1380 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.GPMScript) cvn H.B /ANN pdfmark end 3586 1380 a FK(])125 1488 y(and)h(the)i(output)f(therein.)125 1605 y SDict begin H.S end 125 1605 a 125 1605 a SDict begin 13 H.A end 125 1605 a 125 1605 a SDict begin [/View [/XYZ H.V]/Dest (subsection.8.1) cvn /DEST pdfmark end 125 1605 a 126 x FK(8.1.)46 b FL(Con\014gurations)25 b(with)e(all)h(the)f(isot)m(ypic)i(comp)s (onen)m(ts)f(of)g(m)m(ultiplicit)m(y)f FK(1)p FL(.)46 b FK(This)19 b(case)125 1839 y(is)k(completely)h(analogous)h(to)g(the)g (rank)f(3)g(case,)j(as)e(describ)s(ed)d(in)i(Section)2732 1839 y SDict begin H.S end 2732 1839 a 0 TeXcolorgray FK(7.2)p 0 TeXcolorgray 2848 1780 a SDict begin H.R end 2848 1780 a 2848 1839 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.7.2) cvn H.B /ANN pdfmark end 2848 1839 a FK(.)39 b(Since)23 b(there)i(are)f(no)125 1947 y(isot)m(ypic)e(comp)s(onen)m (ts)i(of)g(higher)e(m)m(ultiplicit)m(y)-8 b(,)22 b(the)i(motiv)m(e)g (of)f(the)h(v)-5 b(ariet)m(y)24 b FC(M)2902 1961 y Fx(\034)2968 1947 y FK(and)f(of)h(the)f(gauge)125 2055 y(group)29 b FC(G)436 2069 y Fx(\034)510 2055 y FK(can)i(b)s(e)f(directly)f (computed)h(using)f(\()1824 2055 y SDict begin H.S end 1824 2055 a 0 TeXcolorgray FK(24)p 0 TeXcolorgray 1916 1996 a SDict begin H.R end 1916 1996 a 1916 2055 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.5.24) cvn H.B /ANN pdfmark end 1916 2055 a FK(\))h(and)g(\()2193 2055 y SDict begin H.S end 2193 2055 a 0 TeXcolorgray FK(25)p 0 TeXcolorgray 2284 1996 a SDict begin H.R end 2284 1996 a 2284 2055 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.5.25) cvn H.B /ANN pdfmark end 2284 2055 a FK(\),)h(whic)m(h)f(are)h(purely)d(com)m (binatorial)125 2163 y(form)m(ulas)d(dep)s(ending)g(on)h(the)h (coincidence)f(of)h(eigen)m(v)-5 b(alues)27 b(on)f(the)h(di\013eren)m (t)g(blo)s(c)m(ks)f(of)h FD(A)g FK(and)f FD(B)5 b FK(.)224 2306 y(There)39 b(are)g(sev)m(en)g(con\014gurations)g(of)g(eigen)m(v)-5 b(alues)38 b(with)g(this)g(prop)s(ert)m(y)-8 b(,)41 b(whose)e(coun)m(t) g(is)f(as)125 2414 y(follo)m(ws.)h(Here)31 b(eigen)m(v)-5 b(alues)30 b(with)g(di\013eren)m(t)f(subindices)f(are)j(distinct)d(n)m (um)m(b)s(ers.)p 0 TeXcolorgray 340 2588 a FC(\017)p 0 TeXcolorgray 42 w FD(\024)e FK(=)f(\(\()p FD(\017)708 2602 y FJ(1)748 2588 y FD(;)15 b(\017)825 2602 y FJ(1)865 2588 y FD(;)g(\017)942 2602 y FJ(1)981 2588 y FD(;)g(\017)1058 2602 y FJ(2)1098 2588 y FK(\))p FD(;)g FK(\()p FD(")1250 2602 y FJ(1)1291 2588 y FD(;)g(")1373 2602 y FJ(2)1413 2588 y FD(;)g(")1495 2602 y FJ(3)1536 2588 y FD(;)g(")1618 2602 y FJ(4)1658 2588 y FK(\)\),)920 2761 y([)p FD(R)1015 2724 y FJ(irr)1014 2784 y Fx(\024)1094 2761 y FK(])26 b(=)14 b FD(q)1274 2724 y FJ(18)1369 2761 y FC(\000)20 b FK(4)15 b FD(q)1564 2724 y FJ(17)1659 2761 y FK(+)20 b(5)15 b FD(q)1854 2724 y FJ(16)1950 2761 y FC(\000)20 b FD(q)2085 2724 y FJ(15)2179 2761 y FC(\000)g FK(3)15 b FD(q)2374 2724 y FJ(14)2470 2761 y FK(+)20 b(3)15 b FD(q)2665 2724 y FJ(13)1235 2913 y FC(\000)20 b FK(5)15 b FD(q)1430 2876 y FJ(12)1526 2913 y FK(+)20 b(7)15 b FD(q)1721 2876 y FJ(11)1816 2913 y FC(\000)20 b FK(2)15 b FD(q)2011 2876 y FJ(10)2106 2913 y FK(+)20 b FD(q)2241 2876 y FJ(9)2301 2913 y FC(\000)g FK(6)15 b FD(q)2496 2876 y FJ(7)2556 2913 y FK(+)20 b(4)15 b FD(q)2751 2876 y FJ(6)2790 2913 y FD(:)p 0 TeXcolorgray 340 3087 a FC(\017)p 0 TeXcolorgray 42 w FD(\024)26 b FK(=)f(\(\()p FD(\017)708 3101 y FJ(1)748 3087 y FD(;)15 b(\017)825 3101 y FJ(1)865 3087 y FD(;)g(\017)942 3101 y FJ(2)981 3087 y FD(;)g(\017)1058 3101 y FJ(2)1098 3087 y FK(\))p FD(;)g FK(\()p FD(")1250 3101 y FJ(1)1291 3087 y FD(;)g(")1373 3101 y FJ(2)1413 3087 y FD(;)g(")1495 3101 y FJ(3)1536 3087 y FD(;)g(")1618 3101 y FJ(4)1658 3087 y FK(\)\),)569 3261 y([)p FD(R)664 3223 y FJ(irr)663 3283 y Fx(\024)743 3261 y FK(])26 b(=)15 b FD(q)924 3223 y FJ(20)1018 3261 y FK(+)20 b(4)15 b FD(q)1213 3223 y FJ(19)1309 3261 y FC(\000)20 b FK(12)15 b FD(q)1549 3223 y FJ(18)1644 3261 y FC(\000)20 b FK(4)15 b FD(q)1839 3223 y FJ(17)1935 3261 y FK(+)20 b(24)15 b FD(q)2175 3223 y FJ(16)2270 3261 y FC(\000)20 b FK(11)15 b FD(q)2510 3223 y FJ(15)2606 3261 y FC(\000)20 b FK(3)15 b FD(q)2801 3223 y FJ(14)2896 3261 y FC(\000)20 b FK(7)15 b FD(q)3091 3223 y FJ(13)885 3413 y FC(\000)20 b FK(6)15 b FD(q)1080 3375 y FJ(12)1175 3413 y FK(+)20 b(25)15 b FD(q)1415 3375 y FJ(11)1511 3413 y FC(\000)20 b FK(3)15 b FD(q)1706 3375 y FJ(10)1801 3413 y FK(+)20 b(11)15 b FD(q)2041 3375 y FJ(9)2101 3413 y FC(\000)20 b FK(19)15 b FD(q)2341 3375 y FJ(8)2402 3413 y FC(\000)20 b FK(18)15 b FD(q)2642 3375 y FJ(7)2702 3413 y FK(+)20 b(18)15 b FD(q)2942 3375 y FJ(6)2983 3413 y FD(:)p 0 TeXcolorgray 340 3586 a FC(\017)p 0 TeXcolorgray 42 w FD(\024)26 b FK(=)f(\(\()p FD(\017)708 3600 y FJ(1)748 3586 y FD(;)15 b(\017)825 3600 y FJ(1)865 3586 y FD(;)g(\017)942 3600 y FJ(2)981 3586 y FD(;)g(\017)1058 3600 y FJ(3)1098 3586 y FK(\))p FD(;)g FK(\()p FD(")1250 3600 y FJ(1)1291 3586 y FD(;)g(")1373 3600 y FJ(2)1413 3586 y FD(;)g(")1495 3600 y FJ(3)1536 3586 y FD(;)g(")1618 3600 y FJ(4)1658 3586 y FK(\)\),)356 3760 y([)p FD(R)451 3722 y FJ(irr)450 3782 y Fx(\024)530 3760 y FK(])25 b(=)15 b FD(q)710 3722 y FJ(22)805 3760 y FK(+)20 b(5)15 b FD(q)1000 3722 y FJ(21)1095 3760 y FK(+)20 b(6)15 b FD(q)1290 3722 y FJ(20)1386 3760 y FC(\000)20 b FK(40)15 b FD(q)1626 3722 y FJ(19)1721 3760 y FC(\000)20 b FK(13)15 b FD(q)1961 3722 y FJ(18)2057 3760 y FK(+)20 b(57)15 b FD(q)2297 3722 y FJ(17)2393 3760 y FK(+)20 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FJ(2)1658 4085 y FK(\)\),)775 4259 y([)p FD(R)870 4222 y FJ(irr)869 4282 y Fx(\024)949 4259 y FK(])25 b(=)15 b FD(q)1129 4222 y FJ(18)1224 4259 y FC(\000)20 b FK(4)15 b FD(q)1419 4222 y FJ(17)1514 4259 y FK(+)20 b(5)15 b FD(q)1709 4222 y FJ(16)1805 4259 y FC(\000)k FD(q)1939 4222 y FJ(15)2034 4259 y FC(\000)h FK(3)15 b FD(q)2229 4222 y FJ(14)1090 4411 y FK(+)20 b(3)15 b FD(q)1285 4374 y FJ(13)1381 4411 y FC(\000)k FK(5)c FD(q)1575 4374 y FJ(12)1671 4411 y FK(+)20 b(7)15 b FD(q)1866 4374 y FJ(11)1961 4411 y FC(\000)20 b FK(2)15 b FD(q)2156 4374 y FJ(10)2251 4411 y FK(+)20 b FD(q)2386 4374 y FJ(9)2446 4411 y FC(\000)g FK(6)15 b FD(q)2641 4374 y FJ(7)2701 4411 y FK(+)20 b(4)15 b FD(q)2896 4374 y FJ(6)2936 4411 y FD(:)p 0 TeXcolorgray 340 4585 a FC(\017)p 0 TeXcolorgray 42 w FD(\024)26 b FK(=)f(\(\()p FD(\017)708 4599 y FJ(1)748 4585 y FD(;)15 b(\017)825 4599 y FJ(2)865 4585 y FD(;)g(\017)942 4599 y FJ(3)981 4585 y FD(;)g(\017)1058 4599 y FJ(4)1098 4585 y FK(\))p FD(;)g FK(\()p FD(")1250 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FK(\))h(and)125 1853 y(\()160 1853 y SDict begin H.S end 160 1853 a 0 TeXcolorgray FK(3)p 0 TeXcolorgray 205 1794 a SDict begin H.R end 205 1794 a 205 1853 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.3) cvn H.B /ANN pdfmark end 205 1853 a FK(\))e(of)g(Section)676 1853 y SDict begin H.S end 676 1853 a 0 TeXcolorgray FK(5)p 0 TeXcolorgray 722 1794 a SDict begin H.R end 722 1794 a 722 1853 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.5) cvn H.B /ANN pdfmark end 722 1853 a FK(.)38 b(As)25 b(alw)m(a)m(ys,)i(the)e(calculation)f(of)h(the)g(motiv)m(e)h(of)f(the)g (gauge)h(group,)f([)p FC(G)3231 1867 y Fx(\034)3275 1853 y FK(],)h(as)f(w)m(ell)125 1961 y(as)36 b(the)h(m)m(ultiplicities)c FD(m)1035 1975 y Fx(\024)1079 1961 y FK(\()p FD(\034)10 b FK(\))37 b(are)g(a)g(purely)e(com)m(binatorial)g(matter.)60 b(Finally)-8 b(,)37 b(the)g(irreducible)125 2070 y(part)30 b([)p FB(M)442 2037 y FJ(irr)442 2092 y Fx(\034)521 2070 y FK(])h(can)f(b)s(e)g(obtained)g(from)g(the)g(results)f(of)i(Section) 2323 2070 y SDict begin H.S end 2323 2070 a 0 TeXcolorgray FK(7)p 0 TeXcolorgray 2369 2011 a SDict begin H.R end 2369 2011 a 2369 2070 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.7) cvn H.B /ANN pdfmark end 2369 2070 a FK(.)224 2213 y(Therefore,)23 b(w)m(e)f(get)g(three)g(con\014guration)f(of)g (eigen)m(v)-5 b(alues,)23 b(whose)e(con)m(tributions)f(are)i(as)f (follo)m(ws:)p 0 TeXcolorgray 340 2399 a FC(\017)p 0 TeXcolorgray 42 w FD(\024)26 b FK(=)f(\(\()p FD(\017)708 2413 y FJ(1)748 2399 y FD(;)15 b(\017)825 2413 y FJ(1)865 2399 y FD(;)g(\017)942 2413 y FJ(2)981 2399 y FD(;)g(\017)1058 2413 y FJ(2)1098 2399 y FK(\))p FD(;)g FK(\()p FD(")1250 2413 y FJ(1)1291 2399 y FD(;)g(")1373 2413 y FJ(1)1413 2399 y FD(;)g(")1495 2413 y FJ(2)1536 2399 y FD(;)g(")1618 2413 y FJ(3)1658 2399 y FK(\)\),)920 2598 y([)p FD(R)1015 2560 y FJ(irr)1014 2620 y Fx(\024)1094 2598 y FK(])26 b(=)14 b FD(q)1274 2560 y FJ(18)1369 2598 y FC(\000)20 b FK(3)15 b FD(q)1564 2560 y FJ(17)1659 2598 y FK(+)20 b(4)15 b FD(q)1854 2560 y FJ(16)1950 2598 y FC(\000)20 b FK(2)15 b FD(q)2145 2560 y FJ(15)2240 2598 y FC(\000)20 b FK(3)15 b FD(q)2435 2560 y FJ(14)2530 2598 y FK(+)20 b(3)15 b FD(q)2725 2560 y FJ(13)1235 2750 y FC(\000)20 b FK(3)15 b FD(q)1430 2713 y FJ(12)1526 2750 y FK(+)20 b(7)15 b FD(q)1721 2713 y FJ(11)1816 2750 y FC(\000)20 b FK(2)15 b FD(q)2011 2713 y FJ(10)2106 2750 y FC(\000)20 b FD(q)2241 2713 y FJ(8)2301 2750 y FC(\000)g FK(5)15 b FD(q)2496 2713 y FJ(7)2556 2750 y FK(+)20 b(4)15 b FD(q)2751 2713 y FJ(6)2790 2750 y FD(:)p 0 TeXcolorgray 340 2949 a FC(\017)p 0 TeXcolorgray 42 w FD(\024)26 b FK(=)f(\(\()p FD(\017)708 2963 y FJ(1)748 2949 y FD(;)15 b(\017)825 2963 y FJ(1)865 2949 y FD(;)g(\017)942 2963 y FJ(2)981 2949 y FD(;)g(\017)1058 2963 y FJ(3)1098 2949 y FK(\))31 b(\()p FD(")1241 2963 y FJ(1)1281 2949 y FD(;)15 b(")1363 2963 y FJ(1)1403 2949 y FD(;)g(")1485 2963 y FJ(2)1525 2949 y FD(;)g(")1607 2963 y FJ(2)1648 2949 y FK(\)\),)920 3148 y([)p FD(R)1015 3110 y FJ(irr)1014 3170 y Fx(\024)1094 3148 y FK(])26 b(=)14 b FD(q)1274 3110 y FJ(18)1369 3148 y FC(\000)20 b FK(3)15 b FD(q)1564 3110 y FJ(17)1659 3148 y FK(+)20 b(4)15 b FD(q)1854 3110 y FJ(16)1950 3148 y FC(\000)20 b FK(2)15 b FD(q)2145 3110 y FJ(15)2240 3148 y FC(\000)20 b FK(3)15 b FD(q)2435 3110 y FJ(14)2530 3148 y FK(+)20 b(3)15 b FD(q)2725 3110 y FJ(13)1235 3300 y FC(\000)20 b FK(3)15 b FD(q)1430 3262 y FJ(12)1526 3300 y FK(+)20 b(7)15 b FD(q)1721 3262 y FJ(11)1816 3300 y FC(\000)20 b FK(2)15 b FD(q)2011 3262 y FJ(10)2106 3300 y FC(\000)20 b FD(q)2241 3262 y FJ(8)2301 3300 y FC(\000)g FK(5)15 b FD(q)2496 3262 y FJ(7)2556 3300 y FK(+)20 b(4)15 b FD(q)2751 3262 y FJ(6)2790 3300 y FD(:)p 0 TeXcolorgray 340 3499 a FC(\017)p 0 TeXcolorgray 42 w FD(\024)26 b FK(=)f(\(\()p FD(\017)708 3513 y FJ(1)748 3499 y FD(;)15 b(\017)825 3513 y FJ(1)865 3499 y FD(;)g(\017)942 3513 y FJ(2)981 3499 y FD(;)g(\017)1058 3513 y FJ(3)1098 3499 y FK(\))p FD(;)g FK(\()p FD(")1250 3513 y FJ(1)1291 3499 y FD(;)g(")1373 3513 y FJ(1)1413 3499 y FD(;)g(")1495 3513 y FJ(2)1536 3499 y FD(;)g(")1618 3513 y FJ(3)1658 3499 y FK(\)\),)569 3698 y([)p FD(R)664 3660 y FJ(irr)663 3720 y Fx(\024)743 3698 y FK(])26 b(=)15 b FD(q)924 3660 y FJ(20)1018 3698 y FK(+)20 b(2)15 b FD(q)1213 3660 y FJ(19)1309 3698 y FC(\000)20 b FK(11)15 b FD(q)1549 3660 y FJ(18)1644 3698 y FK(+)20 b(4)15 b FD(q)1839 3660 y FJ(17)1935 3698 y FK(+)20 b(18)15 b FD(q)2175 3660 y FJ(16)2270 3698 y FC(\000)20 b FK(15)15 b FD(q)2510 3660 y FJ(15)2606 3698 y FC(\000)20 b FK(5)15 b FD(q)2801 3660 y FJ(14)2896 3698 y FC(\000)20 b FK(8)15 b FD(q)3091 3660 y FJ(13)885 3850 y FK(+)20 b(5)15 b FD(q)1080 3812 y FJ(12)1175 3850 y FK(+)20 b(24)15 b FD(q)1415 3812 y FJ(11)1511 3850 y FC(\000)20 b FD(q)1646 3812 y FJ(10)1740 3850 y FK(+)g(4)15 b FD(q)1935 3812 y FJ(9)1995 3850 y FC(\000)20 b FK(24)15 b FD(q)2235 3812 y FJ(8)2296 3850 y FC(\000)20 b FK(11)15 b FD(q)2536 3812 y FJ(7)2596 3850 y FK(+)20 b(17)15 b FD(q)2836 3812 y FJ(6)2876 3850 y FD(:)224 4036 y FK(With)27 b(these)g(results,)g(w)m (e)h(can)g(\014nally)d(compute)i(the)g(c)m(haracter)i(v)-5 b(ariet)m(y)28 b(of)f(irreducible)d(comp)s(o-)125 4144 y(nen)m(ts)h(of)g(torus)g(knots)g(in)f(SL)1134 4158 y FJ(4)1174 4144 y FK(.)39 b(Consider)23 b(the)i(partitions)f FD(\031)2225 4158 y FJ(0)2290 4144 y FK(=)2386 4070 y Ft(\010)2439 4144 y FK(1)2484 4111 y FJ(4)2524 4070 y Ft(\011)2577 4144 y FK(,)i FD(\031)2680 4158 y FJ(1)2744 4144 y FK(=)2840 4070 y Ft(\010)2893 4144 y FK(1)2938 4111 y FJ(2)2978 4144 y FD(;)15 b FK(2)3063 4111 y FJ(1)3104 4070 y Ft(\011)3157 4144 y FK(,)26 b FD(\031)3260 4158 y FJ(2)3325 4144 y FK(=)3421 4070 y Ft(\010)3474 4144 y FK(2)3519 4111 y FJ(2)3558 4070 y Ft(\011)125 4279 y FK(and)j FD(\031)353 4293 y FJ(3)417 4279 y FK(=)513 4206 y Ft(\010)566 4279 y FK(1)611 4246 y FJ(1)651 4279 y FD(;)15 b FK(3)736 4246 y FJ(1)776 4206 y Ft(\011)859 4279 y FK(and,)30 b(giv)m(en)g(t)m(w)m(o)h(of)f(suc)m(h)f(partitions,)g (shorten)g FD(C)2604 4295 y Fx(\031)r(;\031)2710 4276 y Fr(0)2761 4279 y FK(=)c FC(j)p FD(M)2980 4235 y Fx(\031)r(;\031)3086 4212 y Fr(0)2970 4305 y Fx(n;m;)p FJ(4)3155 4279 y FC(j)19 b FK(+)g FC(j)p FD(M)3412 4235 y Fx(\031)3455 4212 y Fr(0)3478 4235 y Fx(;\031)3402 4305 y(n;m;)p FJ(4)3586 4279 y FC(j)125 4399 y FK(if)29 b FD(\031)f FC(6)p FK(=)d FD(\031)439 4366 y Fs(0)493 4399 y FK(and)30 b FD(C)735 4413 y Fx(\031)r(;\031)869 4399 y FK(=)25 b FC(j)p FD(M)1088 4355 y Fx(\031)r(;\031)1078 4425 y(n;m;)p FJ(4)1263 4399 y FC(j)p FK(.)41 b(With)29 b(this)h(notation,)h(the)f(\014nal)f(result) g(is)125 4843 y(\(40\))354 4843 y SDict begin H.S end 354 4843 a 354 4843 a SDict begin 13 H.A end 354 4843 a 354 4843 a SDict begin [/View [/XYZ H.V]/Dest (equation.8.40) cvn /DEST pdfmark end 354 4843 a -229 x FK([)p FB(M)474 4577 y FJ(irr)474 4637 y(4)554 4614 y FK(])c(=)15 b FD(C)755 4628 y Fx(\031)796 4637 y FG(0)830 4628 y Fx(;\031)891 4637 y FG(0)929 4541 y Ft(\000)971 4614 y FD(q)1015 4577 y FJ(9)1074 4614 y FK(+)20 b(6)p FD(q)1254 4577 y FJ(8)1314 4614 y FK(+)g(20)p FD(q)1539 4577 y FJ(7)1599 4614 y FK(+)g(17)p FD(q)1824 4577 y FJ(6)1885 4614 y FC(\000)g FK(98)p FD(q)2110 4577 y FJ(5)2170 4614 y FC(\000)g FK(26)p FD(q)2395 4577 y FJ(4)2455 4614 y FK(+)g(38)p FD(q)2680 4577 y FJ(3)2741 4614 y FK(+)f(126)p FD(q)3010 4577 y FJ(2)3071 4614 y FC(\000)h FK(144)3297 4541 y Ft(\001)695 4767 y FK(+)g FD(C)851 4781 y Fx(\031)892 4790 y FG(0)926 4781 y Fx(;\031)987 4790 y FG(1)1025 4693 y Ft(\000)1067 4767 y FD(q)1111 4729 y FJ(7)1170 4767 y FK(+)g(5)p FD(q)1350 4729 y FJ(6)1410 4767 y FK(+)g(7)p FD(q)1590 4729 y FJ(5)1650 4767 y FC(\000)g FK(34)p FD(q)1875 4729 y FJ(4)1935 4767 y FK(+)g(34)p FD(q)2160 4729 y FJ(2)2220 4767 y FK(+)g(18)p FD(q)k FC(\000)c FK(48)2647 4693 y Ft(\001)695 4919 y FK(+)g FD(C)851 4933 y Fx(\031)892 4942 y FG(0)926 4933 y Fx(;\031)987 4942 y FG(2)1025 4845 y Ft(\000)1067 4919 y FD(q)1111 4881 y FJ(5)1170 4919 y FK(+)g(4)p FD(q)1350 4881 y FJ(4)1410 4919 y FC(\000)g FK(11)p FD(q)1635 4881 y FJ(3)1695 4919 y FK(+)g FD(q)1830 4881 y FJ(2)1890 4919 y FK(+)g(18)p FD(q)k FC(\000)19 b FK(18)2316 4845 y Ft(\001)2379 4919 y FK(+)h FD(C)2535 4933 y Fx(\031)2576 4942 y FG(0)2610 4933 y Fx(;\031)2671 4942 y FG(3)2709 4845 y Ft(\000)2751 4919 y FD(q)2795 4881 y FJ(3)2854 4919 y FC(\000)g FK(4)p FD(q)3034 4881 y FJ(2)3094 4919 y FK(+)g(6)p FD(q)k FC(\000)19 b FK(4)3430 4845 y Ft(\001)695 5071 y FK(+)h FD(C)851 5085 y Fx(\031)892 5094 y FG(1)926 5085 y Fx(;\031)987 5094 y FG(2)1025 4997 y Ft(\000)1067 5071 y FD(q)1111 5033 y FJ(3)1170 5071 y FC(\000)g FK(3)p FD(q)1350 5033 y FJ(2)1410 5071 y FK(+)g(5)p FD(q)j FC(\000)d FK(4)1746 4997 y Ft(\001)1809 5071 y FK(+)g FD(C)1965 5085 y Fx(\031)2006 5094 y FG(1)2040 5085 y Fx(;\031)2101 5094 y FG(1)2139 4997 y Ft(\000)2180 5071 y FD(q)2224 5033 y FJ(5)2284 5071 y FK(+)g(2)p FD(q)2464 5033 y FJ(4)2524 5071 y FC(\000)f FK(10)p FD(q)2748 5033 y FJ(3)2809 5071 y FK(+)h(7)p FD(q)2989 5033 y FJ(2)3049 5071 y FK(+)g(11)p FD(q)k FC(\000)19 b FK(17)3475 4997 y Ft(\001)3518 5071 y FD(:)224 5305 y FK(The)41 b(form)m(ulas)f(for)h FD(C)1017 5319 y Fx(\031)1058 5328 y FG(1)1092 5319 y Fx(;\031)1153 5328 y FG(2)1232 5305 y FK(app)s(ear)g(in)f(Theorem)2056 5305 y SDict begin H.S end 2056 5305 a 0 TeXcolorgray FK(6.8)p 0 TeXcolorgray 2172 5247 a SDict begin H.R end 2172 5247 a 2172 5305 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6.8) cvn H.B /ANN pdfmark end 2172 5305 a 41 w FK(\(and)h(the)h(commen)m(t)g(follo)m(wing)e(it)h(for)125 5413 y FD(r)27 b FK(=)e(4\).)42 b(This)28 b(completes)j(the)g(pro)s(of) e(of)i(Theorem)1945 5413 y SDict begin H.S end 1945 5413 a 0 TeXcolorgray FK(1.1)p 0 TeXcolorgray 2061 5354 a SDict begin H.R end 2061 5354 a 2061 5413 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.1) cvn H.B /ANN pdfmark end 2061 5413 a FK(.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 33 33 TeXDict begin HPSdict begin 33 32 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.33) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 721 66 a FJ(MOTIVE)30 b(OF)g(THE)h(SL)1445 75 y FG(4)1480 66 y FJ(-CHARA)n(CTER)e(V)-8 b(ARIETY)31 b(OF)f(TOR)n(US)h(KNOTS)525 b(33)p 0 TeXcolorgray 125 183 a SDict begin H.S end 125 183 a 125 183 a SDict begin 13 H.A end 125 183 a 125 183 a SDict begin [/View [/XYZ H.V]/Dest (section.9) cvn /DEST pdfmark end 125 183 a 992 266 a FK(9.)46 b FE(Motive)34 b(of)g(the)f(chara)n(cter)f(v)-8 b(arieties)224 463 y FK(T)g(o)31 b(\014nish)d(this)h(w)m(ork,)i(let)f (us)g(compute)h(the)f(motiv)m(e)h(of)g(the)f(c)m(haracter)i(v)-5 b(arieties)1489 624 y FB(X)1554 638 y Fx(r)1618 624 y FK(=)25 b FB(X)p FK(\(\000)1871 638 y Fx(n;m)2000 624 y FD(;)15 b FK(SL)2148 638 y Fx(r)2186 624 y FK(\))p FD(;)125 784 y FK(or)23 b(equiv)-5 b(alen)m(tly)d(,)24 b(of)g(the)g(mo)s(duli)d(spaces)j(of)g(represen)m(tations)f FB(M)2379 798 y Fx(r)2443 784 y FK(=)i FB(M)p FK(\(\000)2726 798 y Fx(n;m)2855 784 y FD(;)15 b FK(SL)3003 798 y Fx(r)3041 784 y FK(\),)25 b(for)f(the)g(torus)125 892 y(knots)29 b(\000)426 906 y Fx(n;;m)605 892 y FK(and)g FD(r)f FK(=)d(4.)41 b(W)-8 b(e)31 b(recall)e(that,)i(for)e(lo)m(w)m(er)h(rank,)g(the)g (motiv)m(e)g(of)g([)p FB(X)2933 906 y FJ(2)2973 892 y FK(])g(app)s(ears)g(in)e([)3495 892 y SDict begin H.S end 3495 892 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 3586 834 a SDict begin H.R end 3586 834 a 3586 892 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 3586 892 a FK(,)125 1000 y(Prop.)i(7.1])h(and)f(the)h(motiv)m(e)g(of) f([)p FB(X)1370 1014 y FJ(3)1410 1000 y FK(])h(is)e(computed)h(in)f([) 2107 1000 y SDict begin H.S end 2107 1000 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 2199 942 a SDict begin H.R end 2199 942 a 2199 1000 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 2199 1000 a FK(,)h(Thm.)g(8.3].)224 1144 y(Recall)g(that)g(the)g(space)g FB(X)1157 1158 y Fx(r)1225 1144 y FK(parametrizes)g(\(isomorphism)d(classes)j(of)7 b(\))31 b(semi-simple)c(represen-)125 1252 y(tations.)40 b(Let)29 b(\005)692 1266 y Fx(r)759 1252 y FK(b)s(e)g(the)g(set)h(of)f (all)f(partitions)f(of)i FD(r)j FK(and)c(let)h(us)f(denote)i(b)m(y)f FD(\031)f FK(=)d FC(f)p FD(r)3095 1211 y Fx(a)3132 1220 y FG(1)3092 1278 y FJ(1)3171 1252 y FD(;)15 b(:)g(:)g(:)i(;)e(r)3417 1219 y Fx(a)3454 1227 y Fq(s)3414 1274 y Fx(s)3492 1252 y FC(g)29 b FK(a)125 1359 y(partition)g(of)h FD(r)s FK(.)40 b(Then)30 b(there)g(is)g(a)g(decomp)s(osition)f(of)i(the)g(c)m (haracter)g(v)-5 b(ariet)m(y)1583 1534 y FB(X)1648 1548 y Fx(r)1712 1534 y FK(=)1846 1447 y Ft(G)1808 1644 y Fx(\031)r Fs(2)p FJ(\005)1951 1652 y Fq(r)2000 1534 y FB(X)2065 1496 y Fx(\031)2065 1556 y(r)2128 1534 y FD(;)125 1786 y FK(where)561 1998 y FB(X)626 1961 y Fx(\031)626 2021 y(r)699 1998 y FK(=)795 1897 y Ft(n)855 1998 y FD(\032)26 b FK(=)1065 1927 y Fx(s)1046 1998 y FC(\010)1024 2064 y Fx(t)p FJ(=1)1154 1870 y Ft(\024)1226 1927 y Fx(a)1263 1935 y Fq(t)1223 1998 y FC(\010)1202 2067 y Fx(l)q FJ(=1)1330 1998 y FD(\032)1377 2013 y Fx(t;l)1448 1870 y Ft(\025)1526 1894 y(\014)1526 1948 y(\014)1526 2003 y(\014)1571 1998 y FD(\032)1618 2013 y Fx(t;l)1715 1998 y FC(2)1811 1974 y FK(~)1801 1998 y FB(X)1866 1961 y FJ(irr)1866 2021 y Fx(r)1898 2029 y Fq(t)1945 1998 y FD(;)15 b FK(det)g FD(\032)26 b FK(=)f(1)2365 1897 y Ft(o)2451 1998 y FC(\032)2589 1885 y Fx(s)2547 1912 y Ft(Y)2547 2106 y Fx(t)p FJ(=1)2678 1998 y FK(Sym)2853 1961 y Fx(a)2890 1969 y Fq(t)2922 1925 y Ft(\000)2974 1974 y FK(~)2964 1998 y FB(X)3029 1961 y FJ(irr)3029 2021 y Fx(r)3061 2029 y Fq(t)3108 1925 y Ft(\001)3150 1998 y FD(:)125 2261 y FK(Here,)32 b(w)m(e)f(set)653 2237 y(~)643 2261 y FB(X)708 2228 y FJ(irr)708 2283 y Fx(r)814 2261 y FK(=)26 b FB(X)976 2228 y FJ(irr)1055 2261 y FK(\(\000)1147 2275 y Fx(n;m)1277 2261 y FD(;)15 b FK(GL)1445 2275 y Fx(r)1483 2261 y FK(\))32 b(and)1737 2237 y(\026)1727 2261 y FB(X)1792 2228 y FJ(irr)1792 2283 y Fx(r)1898 2261 y FK(=)26 b FB(X)2060 2228 y FJ(irr)2139 2261 y FK(\(\000)2231 2275 y Fx(n;m)2361 2261 y FD(;)15 b FK(PGL)2591 2275 y Fx(r)2629 2261 y FK(\),)32 b(as)f(in)f([)2965 2261 y SDict begin H.S end 2965 2261 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 3056 2202 a SDict begin H.R end 3056 2202 a 3056 2261 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 3056 2261 a FK(].)44 b(W)-8 b(e)32 b(also)f(de-)125 2427 y(note)h(b)m(y)456 2299 y Ft(\024)527 2356 y Fx(a)564 2364 y Fq(t)525 2427 y FC(\010)504 2496 y Fx(l)q FJ(=1)631 2427 y FD(\032)678 2442 y Fx(t;l)749 2299 y Ft(\025)829 2427 y FK(the)h(class)e(of)i(the)f (represen)m(tation)g(in)f(the)h(symmetric)f(pro)s(duct)g(Sym)3289 2392 y Fx(a)3326 2400 y Fq(t)3359 2354 y Ft(\000)3410 2404 y FK(~)3400 2427 y FB(X)3465 2394 y FJ(irr)3465 2450 y Fx(r)3497 2458 y Fq(t)3545 2354 y Ft(\001)3586 2427 y FK(.)125 2620 y(Note)24 b(that)h FB(X)593 2572 y Fs(f)p Fx(r)r Fs(g)593 2632 y Fx(r)727 2620 y FK(=)g FB(X)888 2587 y FJ(irr)888 2642 y Fx(r)990 2620 y FK(is)e(the)h(set)g (of)g(irreducible)c(represen)m(tations,)25 b(and)e FB(X)2785 2572 y Fs(f)p FJ(1)2855 2548 y Fq(r)2890 2572 y Fs(g)2785 2632 y Fx(r)2953 2620 y FK(is)g(the)h(set)g(of)g(com-)125 2728 y(pletely)k(reducible)g(represen)m(tations,)i(that)g(is,)f(those)h (whic)m(h)f(are)h(direct)f(sum)g(of)h(one-dimensional)125 2836 y(represen)m(tations.)224 2979 y(In)g(particular,)f(for)h(our)g (rank)g(4)g(case,)i(w)m(e)f(ha)m(v)m(e)125 3161 y(\(41\))780 3161 y SDict begin H.S end 780 3161 a 780 3161 a SDict begin 13 H.A end 780 3161 a 780 3161 a SDict begin [/View [/XYZ H.V]/Dest (equation.9.41) cvn /DEST pdfmark end 780 3161 a FK([)p FB(X)870 3175 y FJ(4)910 3161 y FK(])26 b(=)f([)p FB(X)1147 3124 y Fx(ir)r(r)1147 3184 y FJ(4)1243 3161 y FK(])20 b(+)1379 3088 y Ft(\002)1417 3161 y FB(X)1482 3113 y Fs(f)p FJ(3)p Fx(;)p FJ(1)p Fs(g)1482 3187 y FJ(4)1647 3088 y Ft(\003)1706 3161 y FK(+)1796 3088 y Ft(\002)1834 3161 y FB(X)1899 3113 y Fs(f)p FJ(2)1969 3090 y FG(2)2005 3113 y Fs(g)1899 3187 y FJ(4)2044 3088 y Ft(\003)2102 3161 y FK(+)2193 3088 y Ft(\002)2231 3161 y FB(X)2296 3113 y Fs(f)p FJ(2)p Fx(;)p FJ(1)2421 3090 y FG(2)2457 3113 y Fs(g)2296 3187 y FJ(4)2496 3088 y Ft(\003)2554 3161 y FK(+)2645 3088 y Ft(\002)2683 3161 y FB(X)2748 3113 y Fs(f)p FJ(1)2818 3090 y FG(4)2853 3113 y Fs(g)2748 3187 y FJ(4)2893 3088 y Ft(\003)2931 3161 y FD(:)224 3410 y FK(In)30 b(order)g(to)h(study)e(eac)m(h)j(of)e(the)h(previous)e (strata,)i(w)m(e)g(consider)f(the)g(auxiliary)e(v)-5 b(arieties)934 3616 y(~)924 3640 y FB(X)989 3603 y Fx(\031)989 3663 y(r)1062 3640 y FK(=)1158 3567 y Ft(\010)1211 3640 y FD(\032)25 b FK(=)1421 3569 y Fx(s)1402 3640 y FC(\010)1379 3706 y Fx(t)p FJ(=1)1510 3512 y Ft(\024)1581 3569 y Fx(a)1618 3577 y Fq(t)1579 3640 y FC(\010)1558 3709 y Fx(l)q FJ(=1)1685 3640 y FD(\032)1732 3655 y Fx(t;l)1803 3512 y Ft(\025)1881 3536 y(\014)1881 3590 y(\014)1881 3645 y(\014)1927 3640 y FD(\032)1974 3655 y Fx(t;l)2070 3640 y FC(2)2166 3616 y FK(~)2156 3640 y FB(X)2221 3603 y FJ(irr)2221 3663 y Fx(r)2253 3671 y Fq(t)2325 3567 y Ft(\011)2404 3640 y FK(=)2541 3527 y Fx(s)2500 3554 y Ft(Y)2500 3748 y Fx(t)p FJ(=1)2631 3640 y FK(Sym)2805 3603 y Fx(a)2842 3611 y Fq(t)2875 3567 y Ft(\000)2926 3616 y FK(~)2916 3640 y FB(X)2981 3603 y FJ(irr)2981 3663 y Fx(r)3013 3671 y Fq(t)3060 3567 y Ft(\001)593 3924 y FK(Sym)768 3887 y Fx(a)768 3947 y FJ(0)809 3851 y Ft(\000)861 3901 y FK(~)851 3924 y FB(X)916 3887 y FJ(irr)916 3947 y Fx(r)995 3851 y Ft(\001)1062 3924 y FK(=)1158 3851 y Ft(\010)1211 3924 y FD(\032)g FK(=)1379 3796 y Ft(\024)1464 3853 y Fx(a)1448 3924 y FC(\010)1427 3993 y Fx(l)q FJ(=1)1554 3924 y FD(\032)1601 3939 y Fx(l)1627 3796 y Ft(\025)1706 3820 y(\014)1706 3874 y(\014)1706 3929 y(\014)1751 3924 y FD(\032)1798 3939 y Fx(l)1849 3924 y FC(2)1945 3901 y FK(~)1935 3924 y FB(X)2000 3887 y FJ(irr)2000 3947 y Fx(r)2032 3955 y Fq(t)2079 3924 y FD(;)15 b FK(det)h FD(\032)25 b FK(=)g(1)2474 3851 y Ft(\011)2553 3924 y FC(\032)g FK(Sym)2823 3887 y Fx(a)2865 3851 y Ft(\000)2916 3901 y FK(~)2907 3924 y FB(X)2972 3887 y FJ(irr)2972 3947 y Fx(r)3076 3851 y Ft(\001)3117 3924 y FD(:)224 4184 y FK(If)32 b(w)m(e)h(decomp)s(ose)g(a)f(partition)f FD(\031)h FK(=)1554 4111 y Ft(\010)1607 4184 y FD(r)1651 4144 y Fx(a)1688 4153 y FG(1)1648 4210 y FJ(1)1727 4184 y FD(;)15 b(:)g(:)g(:)i(;)e(r)1973 4151 y Fx(a)2010 4159 y Fq(s)1970 4207 y Fx(s)2048 4111 y Ft(\011)2133 4184 y FK(in)m(to)32 b(the)h(partition)e FD(\031)2914 4151 y Fs(0)2966 4184 y FK(=)3065 4111 y Ft(\010)3118 4184 y FD(r)3162 4144 y Fx(a)3199 4153 y FG(2)3159 4210 y FJ(2)3238 4184 y FD(;)15 b(:)g(:)g(:)i(;)e(r)3484 4151 y Fx(a)3521 4159 y Fq(s)3481 4207 y Fx(s)3558 4111 y Ft(\011)125 4349 y FK(of)37 b FD(r)279 4316 y Fs(0)338 4349 y FK(=)486 4262 y Fx(s)454 4280 y Ft(P)445 4430 y Fx(i)p FJ(=2)575 4349 y FD(a)623 4363 y Fx(i)651 4349 y FD(r)692 4363 y Fx(i)757 4349 y FK(and)941 4275 y Ft(\010)994 4349 y FD(r)1038 4308 y Fx(a)1075 4317 y FG(1)1035 4375 y FJ(1)1113 4275 y Ft(\011)1166 4349 y FK(,)i(w)m(e)f(ha)m(v)m(e)g(a)g (Zariski)d(lo)s(cally)h(trivial)f(\014bration)g FB(X)2990 4316 y Fx(\031)2990 4371 y(r)3074 4349 y FC(!)3212 4325 y FK(~)3202 4349 y FB(X)3267 4316 y Fx(\031)3310 4292 y Fr(0)3267 4377 y Fx(r)3301 4358 y Fr(0)3374 4349 y FK(whose)125 4532 y(\014b)s(er)28 b(is)i(Sym)600 4492 y Fx(a)637 4501 y FG(1)600 4558 y FJ(0)676 4459 y Ft(\000)728 4509 y FK(~)718 4532 y FB(X)783 4499 y FJ(irr)783 4555 y Fx(r)815 4564 y FG(1)862 4459 y Ft(\001)904 4532 y FK(.)40 b(Therefore,)31 b(w)m(e)g(get)g(that)660 4776 y([)p FB(X)750 4739 y Fx(\031)750 4799 y(r)797 4776 y FK(])26 b(=)944 4703 y Ft(\002)982 4776 y FK(Sym)1156 4736 y Fx(a)1193 4745 y FG(1)1156 4802 y FJ(0)1232 4703 y Ft(\000)1284 4753 y FK(~)1274 4776 y FB(X)1339 4739 y FJ(irr)1339 4799 y Fx(r)1371 4808 y FG(1)1418 4703 y Ft(\001\003)1497 4776 y FK([)1532 4753 y(~)1522 4776 y FB(X)1587 4739 y Fx(\031)1630 4715 y Fr(0)1587 4800 y Fx(r)1621 4781 y Fr(0)1658 4776 y FK(])f(=)1804 4703 y Ft(\002)1842 4776 y FK(Sym)2016 4736 y Fx(a)2053 4745 y FG(1)2016 4802 y FJ(0)2092 4703 y Ft(\000)2144 4753 y FK(~)2134 4776 y FB(X)2199 4739 y FJ(irr)2199 4799 y Fx(r)2231 4808 y FG(1)2278 4703 y Ft(\001\003)2415 4663 y Fx(s)2373 4690 y Ft(Y)2373 4884 y Fx(t)p FJ(=2)2504 4703 y Ft(\002)2542 4776 y FK(Sym)2716 4739 y Fx(a)2753 4747 y Fq(t)2786 4703 y Ft(\000)2837 4753 y FK(~)2827 4776 y FB(X)2892 4739 y FJ(irr)2892 4799 y Fx(r)2924 4807 y Fq(t)2972 4703 y Ft(\001)o(\003)3051 4776 y FD(:)125 5027 y SDict begin H.S end 125 5027 a 125 5027 a SDict begin 13 H.A end 125 5027 a 125 5027 a SDict begin [/View [/XYZ H.V]/Dest (theorem.9.1) cvn /DEST pdfmark end 125 5027 a 0 TeXcolorgray FL(Remark)34 b(9.1.)p 0 TeXcolorgray 42 w FK(In)c(man)m(y)g(cases,)i(the)e(previous)f(factors)j(can)e(b)s(e) g(easily)g(computed)g(using)f(kno)m(wn)125 5135 y(results:)p 0 TeXcolorgray 341 5305 a FC(\017)p 0 TeXcolorgray 41 w FK(The)22 b(motiv)m(e)i(of)e(the)h(symmetric)f(pro)s(duct)g(Sym)2084 5269 y Fx(a)2126 5305 y FK(\()p FD(X)7 b FK(\))23 b(can)g(b)s(e)f (computed)h(from)f(the)h(motiv)m(e)427 5413 y(of)f FD(X)28 b FK(b)m(y)21 b(means)h(of)f(the)g(pleth)m(ystic)g(exp)s(onen)m(tial,)h (PExp)f([)2433 5413 y SDict begin H.S end 2433 5413 a 0 TeXcolorgray FK(4)p 0 TeXcolorgray 2478 5354 a SDict begin H.R end 2478 5354 a 2478 5413 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Feng) cvn H.B /ANN pdfmark end 2478 5413 a FK(].)38 b(T)-8 b(o)22 b(b)s(e)e(precise,)j(if)d(the)i(motiv)m (e)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 34 34 TeXDict begin HPSdict begin 34 33 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.34) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(34)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 427 266 a FK(of)30 b FD(X)37 b FK(is)29 b(generated)i(b)m(y)e(the)h(Lefsc)m(hetz) h(motiv)m(e,)g(w)m(e)f(ha)m(v)m(e)h(the)f(follo)m(wing)e([)3052 266 y SDict begin H.S end 3052 266 a 0 TeXcolorgray FK(7)p 0 TeXcolorgray 3097 207 a SDict begin H.R end 3097 207 a 3097 266 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Florentino-Nozad-Zamora:2019a) cvn H.B /ANN pdfmark end 3097 266 a FK(,)i(Prop)s(osition)427 374 y(4.6])1236 531 y Fs(1)1205 558 y Ft(X)1207 752 y Fx(a)p FJ(=0)1352 571 y Ft(\002)1390 644 y FK(Sym)1564 607 y Fx(a)1605 644 y FK(\()p FD(X)7 b FK(\))1757 571 y Ft(\003)1796 644 y FD(z)1842 607 y Fx(a)1910 644 y FK(=)25 b(PExp)2243 571 y Ft(\000)2285 644 y FK([)p FD(X)7 b FK(])p FD(z)2463 571 y Ft(\001)2506 644 y FD(;)427 860 y FK(where,)30 b(at)i(the)e(righ)m(t)g(hand)f(side,)h(w)m(e)h(see)g ([)p FD(X)7 b FK(])p FD(z)35 b FK(as)c(a)g(p)s(olynomial)c(in)i Fz(Z)p FK([)p FD(q)s(;)15 b(z)t FK(].)p 0 TeXcolorgray 341 975 a FC(\017)p 0 TeXcolorgray 41 w FK(If)36 b FD(r)565 989 y FJ(1)639 975 y FK(=)f(1,)j(w)m(e)f(ha)m(v)m(e)h(that)e(Sym)1586 939 y Fx(a)1586 998 y FJ(0)1627 901 y Ft(\000)1679 951 y FK(~)1669 975 y FB(X)1734 942 y FJ(irr)1734 999 y(1)1813 901 y Ft(\001)1891 975 y FK(are)g(the)h(totally)f(reducible)e(represen) m(tations)i(of)427 1100 y(rank)29 b FD(a)p FK(.)40 b(By)29 b([)915 1100 y SDict begin H.S end 915 1100 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 1006 1041 a SDict begin H.R end 1006 1041 a 1006 1100 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 1006 1100 a FK(,)g(Prop.)g(5.2],)i (their)c(motiv)m(es)j(are)f FD(q)2246 1067 y Fx(a)p Fs(\000)p FJ(1)2378 1100 y FK(.)40 b(In)28 b(particular,)2999 1026 y Ft(\002)3036 1100 y FK(Sym)3211 1064 y FJ(1)3211 1122 y(0)3250 1026 y Ft(\000)3302 1076 y FK(~)3292 1100 y FB(X)3357 1067 y FJ(irr)3357 1124 y(1)3436 1026 y Ft(\001\003)3541 1100 y FK(=)427 1144 y Ft(\002)465 1217 y FB(X)530 1184 y FJ(irr)530 1242 y(1)609 1144 y Ft(\003)672 1217 y FK(=)d(1.)224 1417 y(With)34 b(these)g(observ)-5 b(ations)34 b(at)h(hand,)f(w)m(e)h (list)d(the)j(motiv)m(es)f(of)g(eac)m(h)i(of)e(the)g(strata)h(of)g(\() 3378 1417 y SDict begin H.S end 3378 1417 a 0 TeXcolorgray FK(41)p 0 TeXcolorgray 3469 1358 a SDict begin H.R end 3469 1358 a 3469 1417 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.9.41) cvn H.B /ANN pdfmark end 3469 1417 a FK(\))f(of)125 1525 y(the)c(rank)g(4)h(case:)p 0 TeXcolorgray 340 1688 a FC(\017)p 0 TeXcolorgray 42 w FK([)p FB(X)517 1655 y FJ(irr)517 1713 y(4)597 1688 y FK(])25 b(=)g([)p FB(M)863 1655 y FJ(irr)863 1713 y(4)943 1688 y FK(])30 b(is)g(computed)g(in)f (\()1650 1688 y SDict begin H.S end 1650 1688 a 0 TeXcolorgray FK(40)p 0 TeXcolorgray 1741 1630 a SDict begin H.R end 1741 1630 a 1741 1688 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.8.40) cvn H.B /ANN pdfmark end 1741 1688 a FK(\).)p 0 TeXcolorgray 340 1819 a FC(\017)p 0 TeXcolorgray 42 w FK(F)-8 b(or)33 b(the)f(partition)f FC(f)p FK(3)p FD(;)15 b FK(1)p FC(g)p FK(,)35 b(b)m(y)d(the)g(commen)m (ts)h(ab)s(o)m(v)m(e,)g(w)m(e)g(ha)m(v)m(e)2757 1745 y Ft(\002)2795 1819 y FB(X)2860 1771 y Fs(f)p FJ(3)p Fx(;)p FJ(1)p Fs(g)2860 1845 y FJ(4)3025 1745 y Ft(\003)3091 1819 y FK(=)28 b([)3225 1795 y(~)3215 1819 y FB(X)3280 1786 y FJ(irr)3280 1843 y(3)3359 1819 y FK(].)46 b(The)427 1927 y(motiv)m(e)37 b(of)f(this)f(space)h(is)f(giv)m(en)h(in)f([)1748 1927 y SDict begin H.S end 1748 1927 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 1839 1868 a SDict begin H.R end 1839 1868 a 1839 1927 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 1839 1927 a FK(,)j(Cor.)e(10.3])i (\([)2393 1927 y SDict begin H.S end 2393 1927 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 2484 1868 a SDict begin H.R end 2484 1868 a 2484 1927 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 2484 1927 a FK(,)g(Prop.)d(10.1])j(sp)s(eci\014es)d(whic)m(h)427 2035 y(strata)j(corresp)s(ond)d(to)h(irreducible)d(represen)m(tations,) 38 b(hence)f(the)f(p)s(olynomials)d(therein)427 2143 y FD(P)485 2157 y FJ(0)525 2143 y FD(;)15 b(P)623 2157 y FJ(5)663 2143 y FD(;)g(P)761 2157 y FJ(6)826 2143 y FK(should)22 b(b)s(e)i(omitted\).)39 b(Let)26 b(us)d(denote)i(for)g (simplicit)m(y)c FD(P)2788 2157 y FJ(1)2854 2143 y FK(=)j(\()p FD(q)12 b FC(\000)c FK(1\)\()p FD(q)3275 2110 y FJ(4)3324 2143 y FK(+)g(4)p FD(q)3492 2110 y FJ(3)3541 2143 y FC(\000)427 2251 y FK(3)p FD(q)516 2218 y FJ(2)576 2251 y FC(\000)20 b FK(15)p FD(q)k FK(+)c(12\),)32 b FD(P)1153 2265 y FJ(2)1218 2251 y FK(=)25 b(\()p FD(q)e FC(\000)d FK(1\)\()p FD(q)1663 2218 y FJ(4)1724 2251 y FK(+)f(2)p FD(q)1903 2218 y FJ(3)1963 2251 y FC(\000)h FK(3)p FD(q)2143 2218 y FJ(2)2203 2251 y FC(\000)g 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1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.lomune) cvn H.B /ANN pdfmark end 2005 1159 a FK(,)g(Prop.)g(2.6],)i(w)m(e)f(ha)m(v)m (e)945 1270 y Ft(\002)983 1344 y FB(X)1048 1296 y Fs(f)p FJ(2)1118 1273 y FG(2)1153 1296 y Fs(g)1048 1370 y FJ(4)1193 1270 y Ft(\003)1256 1344 y FK(=)25 b([)p Fz(C)1437 1307 y Fs(\003)1482 1344 y FK(])1507 1307 y FJ(+)1586 1344 y FC(\001)c FK([\()1702 1320 y(\026)1692 1344 y FB(X)1757 1307 y FJ(irr)1757 1367 y(2)1837 1344 y FK(\))1872 1307 y FJ(2)1911 1344 y FK(])1936 1307 y FJ(+)2016 1344 y FK(+)f([)p Fz(C)2192 1307 y Fs(\003)2237 1344 y FK(])2262 1307 y Fs(\000)2342 1344 y FC(\001)g FK([\()2457 1320 y(\026)2447 1344 y FB(X)2512 1307 y FJ(irr)2512 1367 y(2)2592 1344 y FK(\))2627 1307 y FJ(2)2667 1344 y FK(])2692 1307 y Fs(\000)2766 1344 y FD(:)427 1508 y FK(Here,)30 b(for)f FD(X)36 b FK(an)29 b(algebraic)f(v)-5 b(ariet)m(y)30 b(with)d(a)i Fz(Z)2066 1522 y FJ(2)2101 1508 y FK(-action,)h(w)m(e)g (denote)f([)p FD(X)7 b FK(])2984 1475 y FJ(+)3069 1508 y FK(=)25 b([)p FD(X=)p Fz(Z)3375 1522 y FJ(2)3411 1508 y FK(])k(and)427 1616 y([)p FD(X)7 b FK(])559 1583 y Fs(\000)645 1616 y FK(=)24 b([)p FD(X)7 b FK(])22 b FC(\000)e FK([)p FD(X)7 b FK(])1117 1583 y FJ(+)1177 1616 y FK(.)40 b(By)31 b([)1410 1616 y SDict begin H.S end 1410 1616 a 0 TeXcolorgray FK(22)p 0 TeXcolorgray 1501 1557 a SDict begin H.R end 1501 1557 a 1501 1616 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.lomune) cvn H.B /ANN pdfmark end 1501 1616 a FK(,)g(Eqn.)f(\(1\)],)h(w)m(e)g(ha)m(v)m(e)h([)p Fz(C)2398 1583 y Fs(\003)2443 1616 y FK(])2468 1583 y FJ(+)2553 1616 y FK(=)25 b FD(q)33 b FK(and)d([)p Fz(C)2985 1583 y Fs(\003)3030 1616 y FK(])3055 1583 y Fs(\000)3140 1616 y FK(=)25 b FC(\000)p FK(1.)527 1724 y(The)30 b(motiv)m(e)h(of) 1125 1701 y(\026)1115 1724 y FB(X)1180 1691 y FJ(irr)1180 1749 y(2)1290 1724 y FK(is)e(pro)m(vided)g(in)g([)1886 1724 y SDict begin H.S end 1886 1724 a 0 TeXcolorgray FK(28)p 0 TeXcolorgray 1977 1666 a SDict begin H.R end 1977 1666 a 1977 1724 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.MP) cvn H.B /ANN pdfmark end 1977 1724 a FK(,)i(Prop.)f(7.2])i (and)d(it)h(is)g(giv)m(en)g(b)m(y)734 1967 y([)769 1943 y(\026)759 1967 y FB(X)824 1929 y FJ(irr)824 1989 y(2)903 1967 y FK(])25 b(=)1049 1811 y Ft(\()1174 1867 y FJ(\()p Fx(n)p Fs(\000)p FJ(1\)\()p Fx(m)p Fs(\000)p FJ(1\))p 1174 1891 396 4 v 1354 1943 a(4)1579 1911 y FK(\()p FD(q)f FC(\000)c FK(2\))638 b(if)30 b FD(n;)15 b(m)30 b FK(o)s(dd,)1174 1994 y FJ(\()p Fx(n)p Fs(\000)p FJ(2\)\()p Fx(m)p Fs(\000)p FJ(1\))p 1174 2018 V 1354 2070 a(4)1579 2038 y FK(\()p FD(q)24 b FC(\000)c FK(2\))h(+)1972 2003 y Fx(m)p Fs(\000)p FJ(1)p 1972 2018 153 4 v 2030 2070 a(2)2134 2038 y FK(\()p FD(q)j FC(\000)19 b FK(1\))84 b(if)30 b FD(n)g FK(ev)m(en.)527 2207 y(By)h([)695 2207 y SDict begin H.S end 695 2207 a 0 TeXcolorgray FK(12)p 0 TeXcolorgray 786 2149 a SDict begin H.R end 786 2149 a 786 2207 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.GLM) cvn H.B /ANN pdfmark end 786 2207 a FK(],)g(the)f(symmetric)g(pro)s(duct)f(is)h(an)g(op)s(eration)g(on)g (the)h(Grothendiec)m(k)f(ring.)40 b(By)427 2315 y(Remark)773 2315 y SDict begin H.S end 773 2315 a 0 TeXcolorgray FK(9.1)p 0 TeXcolorgray 889 2257 a SDict begin H.R end 889 2257 a 889 2315 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.9.1) cvn H.B /ANN pdfmark end 889 2315 a FK(,)34 b(if)e(w)m(e)i(see) g([)p FD(X)7 b FK(])34 b(as)f(a)h(p)s(olynomial)d(in)h FD(q)k FK(denoted)d(b)m(y)g([)p FD(X)7 b FK(])16 b(\()p FD(q)s FK(\),)35 b(w)m(e)f(ha)m(v)m(e)g(the)427 2423 y(form)m(ula)1083 2564 y(Sym)1257 2527 y FJ(2)1297 2564 y FK(\([)p FD(X)7 b FK(])16 b(\()p FD(q)s FK(\)\))26 b(=)1761 2503 y(1)p 1761 2543 46 4 v 1761 2627 a(2)1832 2564 y([)p FD(X)7 b FK(])16 b(\()p FD(q)2059 2527 y FJ(2)2099 2564 y FK(\))k(+)2255 2503 y(1)p 2255 2543 V 2255 2627 a(2)2326 2564 y([)p FD(X)7 b FK(])16 b(\()p FD(q)s FK(\))2588 2527 y FJ(2)2628 2564 y FD(;)427 2730 y FK(Hence)31 b(for)g FD(n;)15 b(m)30 b FK(o)s(dd)f(w)m(e)i(ha)m(v)m(e)268 2870 y Ft(\002)306 2943 y FK(\()351 2920 y(\026)341 2943 y FB(X)406 2906 y FJ(irr)406 2966 y(2)486 2943 y FK(\))521 2906 y FJ(2)561 2870 y Ft(\003)599 2892 y FJ(+)683 2943 y FK(=)779 2870 y Ft(\002)817 2943 y FK(Sym)991 2906 y FJ(2)1030 2943 y FK(\()1075 2920 y(\026)1065 2943 y FB(X)1130 2906 y FJ(irr)1130 2966 y(2)1210 2943 y FK(\))1245 2870 y Ft(\003)1308 2943 y FK(=)1414 2882 y(\()p FD(m)21 b FC(\000)f FK(1\))1721 2849 y FJ(2)1761 2882 y FK(\()p FD(n)g FC(\000)g FK(1\))2042 2849 y FJ(2)p 1414 2922 668 4 v 1703 3006 a FK(32)2092 2943 y(\()p FD(q)j FC(\000)d FK(2\))2362 2906 y FJ(2)2423 2943 y FK(+)2524 2882 y(\()p FD(m)g FC(\000)g FK(1\)\()p FD(n)h FC(\000)f FK(1\))p 2524 2922 589 4 v 2795 3006 a(8)3123 2943 y(\()p FD(q)3202 2906 y FJ(2)3261 2943 y FC(\000)g FK(2\))p FD(;)268 3058 y Ft(\002)306 3132 y FK(\()351 3108 y(\026)341 3132 y FB(X)406 3095 y FJ(irr)406 3155 y(2)486 3132 y FK(\))521 3095 y FJ(2)561 3058 y Ft(\003)599 3081 y Fs(\000)683 3132 y FK(=)779 3058 y Ft(\002)817 3132 y FK(\()862 3108 y(\026)852 3132 y FB(X)917 3095 y FJ(irr)917 3155 y(2)996 3132 y FK(\))1031 3095 y FJ(2)1071 3058 y Ft(\003)1129 3132 y FC(\000)1220 3058 y Ft(\002)1258 3132 y FK(\()1303 3108 y(\026)1293 3132 y FB(X)1358 3095 y FJ(irr)1358 3155 y(2)1437 3132 y FK(\))1472 3095 y FJ(2)1512 3058 y Ft(\003)1550 3081 y FJ(+)683 3341 y FK(=)25 b FC(\000)860 3280 y FK(\()p FD(n)20 b FC(\000)f FK(1\))1140 3247 y FJ(2)1181 3280 y FK(\()p FD(m)h FC(\000)g FK(1\))1487 3247 y FJ(2)p 860 3320 668 4 v 1148 3404 a FK(32)1537 3341 y(\()p FD(q)k FC(\000)c FK(2\))1808 3304 y FJ(2)1868 3341 y FK(+)1969 3280 y(\()p FD(n)g FC(\000)g FK(1\)\()p FD(m)h FC(\000)f FK(1\))p 1969 3320 589 4 v 2241 3404 a(8)2568 3341 y(\()p FC(\000)p FD(q)2718 3304 y FJ(2)2777 3341 y FK(+)g(2)p FD(q)k FC(\000)c FK(2\))p FD(;)372 3503 y Ft(\002)410 3576 y FB(X)475 3528 y Fs(f)p FJ(2)545 3505 y FG(2)580 3528 y Fs(g)475 3602 y FJ(4)620 3503 y Ft(\003)683 3576 y FK(=)789 3515 y(\()p FD(m)g FC(\000)g FK(1\))1095 3482 y FJ(2)1135 3515 y FK(\()p FD(n)g FC(\000)g FK(1\))1416 3482 y FJ(2)p 789 3555 668 4 v 1077 3639 a FK(32)1467 3576 y(\()p FD(q)1546 3539 y FJ(3)1605 3576 y FC(\000)g FK(3)p FD(q)1785 3539 y FJ(2)1845 3576 y FK(+)g(4\))h(+)2138 3515 y(\()p FD(m)f FC(\000)g FK(1\)\()p FD(n)h FC(\000)f FK(1\))p 2138 3555 589 4 v 2410 3639 a(8)2737 3576 y(\()p FD(q)2816 3539 y FJ(3)2876 3576 y FK(+)g FD(q)3011 3539 y FJ(2)3070 3576 y FC(\000)g FK(4)p FD(q)k FK(+)19 b(2\))p FD(;)427 3760 y FK(and)30 b(for)g FD(n)g FK(ev)m(en)h(and)f FD(m)g FK(o)s(dd,)245 3901 y Ft(\002)282 3974 y FK(\()327 3951 y(\026)317 3974 y FB(X)382 3937 y FJ(irr)382 3997 y(2)462 3974 y FK(\))497 3937 y FJ(2)537 3901 y Ft(\003)575 3923 y FJ(+)659 3974 y FK(=)745 3901 y Ft(\002)783 3974 y FK(Sym)957 3937 y FJ(2)996 3974 y FK(\()1041 3951 y(\026)1031 3974 y FB(X)1096 3937 y FJ(irr)1096 3997 y(2)1176 3974 y FK(\))1211 3901 y Ft(\003)1274 3974 y FK(=)1380 3913 y(\()p FD(m)21 b FC(\000)f FK(1\))1687 3880 y FJ(2)1727 3913 y FK(\()p FD(n)g FC(\000)g FK(1\))2008 3880 y FJ(2)p 1380 3953 668 4 v 1669 4037 a FK(32)2058 3974 y(\()p FD(q)j FC(\000)d FK(2\))2328 3937 y FJ(2)2389 3974 y FK(+)2490 3913 y(\()p FD(m)g FC(\000)g FK(1\)\()p FD(n)h FC(\000)f FK(1\))p 2490 3953 589 4 v 2761 4037 a(8)3089 3974 y(\()p FD(q)3168 3937 y FJ(2)3227 3974 y FC(\000)g FK(2\))750 4209 y(+)851 4148 y(\()p FD(m)g FC(\000)g FK(1\))1157 4115 y FJ(2)p 851 4188 347 4 v 1001 4272 a FK(8)1207 4209 y(\()p FD(q)k FC(\000)c FK(1\))1478 4172 y FJ(2)1538 4209 y FK(+)1639 4148 y FD(m)g FC(\000)g FK(1)p 1639 4188 237 4 v 1734 4272 a(4)1885 4209 y(\()p FD(q)1964 4172 y FJ(2)2024 4209 y FC(\000)g FK(1\))h(+)2317 4148 y(\()p FD(n)f FC(\000)g FK(1\)\()p FD(m)h FC(\000)f FK(1\))2905 4115 y FJ(2)p 2317 4188 629 4 v 2608 4272 a FK(8)2955 4209 y(\()p FD(q)3034 4172 y FJ(2)3094 4209 y FC(\000)g FK(3)p FD(q)j FK(+)d(2\))p FD(;)245 4371 y Ft(\002)282 4444 y FK(\()327 4420 y(\026)317 4444 y FB(X)382 4407 y FJ(irr)382 4467 y(2)462 4444 y FK(\))497 4407 y FJ(2)537 4371 y Ft(\003)575 4393 y Fs(\000)659 4444 y FK(=)g FC(\000)851 4383 y FK(\()p FD(n)g FC(\000)g FK(1\))1132 4350 y FJ(2)1172 4383 y FK(\()p FD(m)g FC(\000)g FK(1\))1478 4350 y FJ(2)p 851 4423 668 4 v 1139 4506 a FK(32)1528 4444 y(\()p FD(q)k FC(\000)c FK(2\))1799 4407 y FJ(2)1859 4444 y FK(+)1960 4383 y(\()p FD(n)g FC(\000)g FK(1\)\()p FD(m)h FC(\000)f FK(1\))p 1960 4423 589 4 v 2232 4506 a(8)2559 4444 y(\()p FC(\000)p FD(q)2709 4407 y FJ(2)2769 4444 y FK(+)f(2)p FD(q)24 b FC(\000)c FK(2\))750 4679 y FC(\000)851 4618 y FK(\()p FD(m)g FC(\000)g FK(1\))1157 4585 y FJ(2)p 851 4658 347 4 v 1001 4741 a FK(8)1207 4679 y(\()p FD(q)k FC(\000)c FK(1\))1478 4642 y FJ(2)1538 4679 y FC(\000)1639 4618 y FD(m)g FC(\000)g FK(1)p 1639 4658 237 4 v 1734 4741 a(4)1885 4679 y(\()p FD(q)k 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347 4 v 1001 5211 a FK(8)1207 5149 y(\()p FD(q)1286 5111 y FJ(3)1346 5149 y FC(\000)g FD(q)1481 5111 y FJ(2)1540 5149 y FC(\000)g FD(q)j FK(+)d(1\))h(+)1988 5087 y FD(m)f FC(\000)g FK(1)p 1988 5128 237 4 v 2084 5211 a(4)2235 5149 y(\()p FD(q)2314 5111 y FJ(3)2373 5149 y FK(+)g FD(q)2508 5111 y FJ(2)2568 5149 y FC(\000)g FK(3)p FD(q)j FK(+)d(1\))750 5384 y(+)851 5322 y(\()p FD(n)g FC(\000)g FK(1\)\()p FD(m)h FC(\000)f FK(1\))1439 5289 y FJ(2)p 851 5363 629 4 v 1142 5446 a FK(8)1489 5384 y(\()p FD(q)1568 5346 y FJ(3)1628 5384 y FC(\000)g FK(2)p FD(q)1808 5346 y FJ(2)1868 5384 y FC(\000)g FD(q)j FK(+)d(2\))p FD(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 36 36 TeXDict begin HPSdict begin 36 35 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 116 -8 a SDict begin H.S end 116 -8 a 116 -8 a SDict begin H.R end 116 -8 a 116 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.36) cvn /DEST pdfmark end 116 -8 a 0 TeXcolorgray 125 66 a FJ(36)1195 50 y(\023)1186 66 y(A.)23 b(GONZ)1497 50 y(\023)1488 66 y(ALEZ-PRIETO)h(AND)f(V.)g(MU) 2409 50 y(~)2400 66 y(NOZ)p 0 TeXcolorgray 0 TeXcolorgray 340 266 a FC(\017)p 0 TeXcolorgray 42 w FK(F)-8 b(or)31 b(the)g(partition)1126 192 y Ft(\010)1179 266 y FK(2)p FD(;)15 b FK(1)1309 233 y FJ(2)1350 192 y Ft(\011)1403 266 y FK(,)30 b(w)m(e)h(ha)m(v)m(e)h(that)1121 385 y Ft(\002)1158 459 y FB(X)1223 411 y Fs(f)p FJ(2)p Fx(;)p FJ(1)1348 388 y FG(2)1384 411 y Fs(g)1223 485 y FJ(4)1423 385 y Ft(\003)1486 459 y FK(=)1582 385 y Ft(\002)1620 459 y FK(Sym)1795 421 y FJ(2)1795 481 y(0)1834 385 y Ft(\000)1886 435 y FK(~)1876 459 y FB(X)1941 421 y FJ(irr)1941 481 y(1)2020 385 y Ft(\001)o(\003)2114 459 y FK([)2149 435 y(~)2139 459 y FB(X)2204 473 y FJ(2)2245 459 y FK(])25 b(=)g FD(q)s FK([)2470 435 y(~)2460 459 y 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pdfmark end 1603 1241 a FE(References)125 1295 y SDict begin H.S end 125 1295 a 125 1295 a SDict begin 11 H.A end 125 1295 a 125 1295 a SDict begin [/View [/XYZ H.V]/Dest (cite.Corlette:1988) cvn /DEST pdfmark end 125 1295 a 0 TeXcolorgray 163 1386 a FI([1])p 0 TeXcolorgray 39 w(K.)26 b(Corlette,)h Fy(Flat)h FF(G)p Fy(-bund)t(les)h(with)f(c)l(anonic)l(al)g(metrics)p FI(,)f(J.)f(Di\013.)g(Geom.)g Fd(28)g FI(\(1988\))h(361{382.)125 1405 y SDict begin H.S end 125 1405 a 125 1405 a SDict begin 11 H.A end 125 1405 a 125 1405 a SDict begin [/View [/XYZ H.V]/Dest (cite.CS) cvn /DEST pdfmark end 125 1405 a 0 TeXcolorgray 163 1477 a FI([2])p 0 TeXcolorgray 39 w(M.)35 b(Culler)h(and)e(P)-6 b(.)34 b(Shalen,)j Fy(V)-6 b(arieties)37 b(of)e(gr)l(oup)i(r)l(epr)l(esentations)i(and)d (splitting)g(of)f FI(3)p Fy(-manifolds)p FI(,)j(Annals)282 1569 y(Math.)27 b(\(2\))e Fd(117)i FI(\(1983\))f(109{146.)125 1587 y SDict begin H.S end 125 1587 a 125 1587 a SDict begin 11 H.A end 125 1587 a 125 1587 a SDict 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end 125 1953 a 125 1953 a SDict begin [/View [/XYZ H.V]/Dest (cite.Florentino-Lawton:2012) cvn /DEST pdfmark end 125 1953 a 0 TeXcolorgray 163 2025 a FI([5])p 0 TeXcolorgray 39 w(C.)h(Floren)n(tino)g(and)f(S.)g(La)n(wton,)h Fy(Singularities)h(of)g(fr)l(e)l(e)g(gr)l(oup)h(char)l(acter)h (varieties)p FI(,)e(P)n(ac.)f(J.)g(Math.)g(\(1\))f Fd(260)282 2117 y FI(\(2012\))g(149{179.)125 2135 y SDict begin H.S end 125 2135 a 125 2135 a SDict begin 11 H.A end 125 2135 a 125 2135 a SDict begin [/View [/XYZ H.V]/Dest (cite.Florentino-Nozad-Zamora:2019) cvn /DEST pdfmark end 125 2135 a 0 TeXcolorgray 163 2208 a FI([6])p 0 TeXcolorgray 39 w(C.)32 b(Floren)n(tino,)g(A.)f (Nozad)f(and)g(A.)h(Zamora,)h FF(E)t Fy(-p)l(olynomials)g(of)g FI(SL)2408 2216 y Fq(n)2451 2208 y Fy(-)g(and)h FI(PGL)2824 2216 y Fq(n)2867 2208 y Fy(-char)l(acter)i(varieties)e(of)282 2299 y(fr)l(e)l(e)28 b(gr)l(oups)p FI(,)g(J.)e(Geom.)g(Ph)n(ys.)g Fd(161)g FI(\(2021\).)125 2318 y SDict begin H.S end 125 2318 a 125 2318 a SDict begin 11 H.A end 125 2318 a 125 2318 a SDict begin [/View [/XYZ H.V]/Dest (cite.Florentino-Nozad-Zamora:2019a) cvn /DEST pdfmark end 125 2318 a 0 TeXcolorgray 163 2391 a FI([7])p 0 TeXcolorgray 39 w(C.)38 b(Floren)n(tino,)j(A.)c (Nozad)g(and)f(A.)h(Zamora,)j Fy(Gener)l(ating)g(series)f(for)f(the)h FF(E)t Fy(-p)l(olynomials)f(of)g FI(GL\()p FF(n;)13 b Fc(C)g FI(\))p Fy(-)282 2482 y(char)l(acter)31 b(varieties)p FI(,)c(to)f(app)r(ear)g(in)f(Math.)i(Nac)n(hr.,)1882 2482 y SDict begin H.S end 1882 2482 a 0 TeXcolorgray FI(arXiv:1912.05852)p 0 TeXcolorgray 2459 2431 a SDict begin H.R end 2459 2431 a 2459 2482 a SDict begin [/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 1]/Action <</Subtype/URI/URI(http://arxiv.org/abs/1912.05852)>>/Subtype /Link H.B /ANN pdfmark end 2459 2482 a FI(.)125 2497 y SDict begin H.S end 125 2497 a 125 2497 a SDict begin 11 H.A end 125 2497 a 125 2497 a SDict begin [/View [/XYZ H.V]/Dest (cite.GP:2018) cvn /DEST pdfmark end 125 2497 a 0 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