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index round cvi 8 index round cvi sub 4 mul 1 add def /pgfcircx pgfstartx 9 index pgfdiffx pgfdomb div mul add def /pgfcircy pgfstarty 9 index pgfdiffy pgfdomb div mul add def /pgfcircxe pgfstartx 8 index pgfdiffx pgfdomb div mul add def /pgfcircye pgfstarty 8 index pgfdiffy pgfdomb div mul add def /pgfxstep pgfcircxe pgfcircx sub pgfdiff div def /pgfystep pgfcircye pgfcircy sub pgfdiff div def 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 2 index 6 index sub pgfdiff div 8 index 8 index 8 index 13 index pgfdiff { 3 index 3 index 3 index setrgbcolor pgfcircx pgfcircy 2 index 0 360 arc closepath stroke 4 -1 roll 6 index add 4 -1 roll 5 index add 4 -1 roll 4 index add 4 -1 roll .25 sub /pgfcircx pgfcircx pgfxstep add def /pgfcircy pgfcircy pgfystep add def } repeat mark 14 1 roll cleartomark exch pop }bind def /pgfHcmyk{ /pgfheight exch def 0.75 setlinewidth [] 0 setdash /pgfshade {pgfAcmyk} def /pgfdir { dup 0 moveto dup 6 index lineto } bind def} bind def /pgfVcmyk{ 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newpath}bind def} ifelse}bind def /pgffoa 1 def /pgfsoa 1 def /.pushpdf14devicefilter where {pop [userdict /bop-hook known {userdict /bop-hook get aload pop} if {0 .pushpdf14devicefilter} aload pop] cvx userdict exch /bop-hook exch put [userdict /eop-hook known {userdict /eop-hook get aload pop} if {.poppdf14devicefilter} aload pop] cvx userdict exch /eop-hook exch put} if systemdict /pdfmark known not {userdict /pdfmark systemdict /cleartomark get put} if end /pgfwritesamplecmyk { 4 index 0 5 index pgfcheckcolorrange 255 mul round cvi put 4 index 1 4 index pgfcheckcolorrange 255 mul round cvi put 4 index 2 3 index pgfcheckcolorrange 255 mul round cvi put 4 index 3 2 index pgfcheckcolorrange 255 mul round cvi put pop pop pop pop } bind def /pgfwritesamplergb { 3 index 0 4 index pgfcheckcolorrange 255 mul round cvi put 3 index 1 3 index pgfcheckcolorrange 255 mul round cvi put 3 index 2 2 index pgfcheckcolorrange 255 mul round cvi put pop pop pop } bind def /pgfwritesamplegray { pgfcheckcolorrange 16777215 mul round cvi 1 index 0 2 index -16 bitshift put 1 index 1 2 index 65535 and -8 bitshift put 1 index 2 2 index 255 and put pop } bind def /pgfcheckcolorrange { dup 0.0 lt {pop 0.0} if dup 1.0 gt {pop 1.0} if } bind def /pgfchanneldepthcmyk 8 def /pgfchanneldepthrgb 8 def /pgfchanneldepthgray 24 def /pgfcolorsamplecmyk 4 string def /pgfcolorsamplergb 3 string def /pgfcolorsamplegray 3 string def /pgfrangecmyk [0 1 0 1 0 1 0 1] def /pgfrangergb [0 1 0 1 0 1] def /pgfrangegray [0 1] def /pgf1{gsave exec 1.0 pgfw 2.00002 0.0 moveto -6.00006 4.00005 lineto -3.00003 0.0 lineto -6.00006 -4.00005 lineto pgffill grestore} bind def /pgf2{gsave exec 1.0 pgfw 0.8 pgfw [ ] 0.0 setdash 1 setlinecap 1 setlinejoin -3.00003 4.00005 moveto -2.75002 2.50002 0.0 0.24998 0.75 0.0 curveto 0.0 -0.24998 -2.75002 -2.50002 -3.00003 -4.00005 curveto pgfstr grestore} bind def /pgf3{gsave exec 1.0 pgfw [ ] 0.0 setdash 0.0 -5.00005 moveto 0.0 5.00005 lineto pgfstr grestore} bind def /pgf4{gsave exec 1.0 pgfw [ ] 0.0 setdash -3.00003 -5.00005 moveto 0.0 -5.00005 lineto 0.0 5.00005 lineto -3.00003 5.00005 lineto pgfstr grestore} bind def /pgf5{gsave exec 1.0 pgfw [ ] 0.0 setdash -2.00002 -5.00005 moveto 1.0 -3.00003 1.0 3.00003 -2.00002 5.00005 curveto pgfstr grestore} bind def /pgf6{gsave exec 1.0 pgfw [ ] 0.0 setdash -4.50003 -5.00005 moveto 0.49998 0.0 lineto -4.50003 5.00005 lineto pgfstr grestore} bind def /pgf7{gsave exec 1.0 pgfw -2.50002 0.0 translate [ ] 0.0 setdash 3.00003 0.0 moveto 3.00003 1.65689 1.65689 3.00003 0.0 3.00003 curveto -1.65689 3.00003 -3.00003 1.65689 -3.00003 0.0 curveto -3.00003 -1.65689 -1.65689 -3.00003 0.0 -3.00003 curveto 1.65689 -3.00003 3.00003 -1.65689 3.00003 0.0 curveto closepath gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath grestore} bind def /pgf8{gsave exec 1.0 pgfw [ ] 0.0 setdash 1.0 0.0 moveto -5.00005 3.00003 lineto -11.00012 0.0 lineto -5.00005 -3.00003 lineto closepath gsave pgffc pgffill grestore 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3070 3090 a SDict begin H.R end 3070 3090 a 3070 3152 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.Ram) cvn H.B /ANN pdfmark end 3070 3152 a 0 TeXcolorgray Ft(])h(pro)m(v)m(ed)f(that)h (the)75 3260 y(F)-8 b(rob)s(enius)27 b(c)m(haracter)j(form)m(ula)e(for) g Fy(H)1432 3274 y Fq(n)1478 3260 y Ft(\()p Fy(q)s Ft(\))h(in)f(terms)g (of)h(the)f(one-ro)m(w)i(Hall-Lilltew)m(o)s(o)s(d)25 b(functions)j(and)75 3370 y(Sc)m(h)m(ur)j(functions)g(via)g(the)h(quan) m(tum)g(Sc)m(h)m(ur-W)-8 b(eyl)32 b(dualit)m(y)-8 b(.)31 b(An)m(y)h(irreducible)c(c)m(haracter)34 b Fy(\037)3331 3337 y Fq(\025)3408 3370 y Ft(of)e Fy(H)3589 3384 y Fq(n)3635 3370 y Ft(\()p Fy(q)s Ft(\))75 3478 y(is)k(completely)g(determined)f(b) m(y)i(its)f(v)-5 b(alues)36 b(on)h(the)f(elemen)m(ts)h Fy(T)2386 3492 y Fq(\015)2422 3500 y Fu(\026)2469 3478 y Fy(;)15 b(\026)35 b Fm(`)h Fy(n)g Ft(\(see)h([)p 0 1 0 0 TeXcolorcmyk 2995 3478 a SDict begin H.S end 2995 3478 a 0 1 0 0 TeXcolorcmyk Ft(C86)p 0 1 0 0 TeXcolorcmyk 3152 3416 a SDict begin H.R end 3152 3416 a 3152 3478 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.Ca) cvn H.B /ANN pdfmark end 3152 3478 a 0 TeXcolorgray Ft(,)p 0 1 0 0 TeXcolorcmyk 3214 3478 a SDict begin H.S end 3214 3478 a 0 1 0 0 TeXcolorcmyk Ft(R91)p 0 1 0 0 TeXcolorcmyk 3372 3416 a SDict begin H.R end 3372 3416 a 3372 3478 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.Ram) cvn H.B /ANN pdfmark end 3372 3478 a 0 TeXcolorgray Ft(]\).)g(Let)h(us)75 3597 y(denote)c(the)f(v)-5 b(alue)33 b(of)g(irreducible)d(c)m(haracter) 35 b Fy(\037)1784 3564 y Fq(\025)1862 3597 y Ft(on)e(the)h(elemen)m(t)g Fy(T)2543 3611 y Fq(\015)2579 3619 y Fu(\026)2658 3597 y Ft(b)m(y)f Fy(\037)2844 3564 y Fq(\025)2889 3597 y Ft(\()p Fy(T)2977 3611 y Fq(\015)3013 3619 y Fu(\026)3060 3597 y Ft(\),)g(abbreviated)g(b)m(y)75 3716 y Fy(\037)132 3683 y Fq(\025)132 3739 y(\026)178 3716 y Ft(.)175 3828 y(In)m(tro)s(duce)c(the)i(mo)s(di\014cation)36 b(~)-52 b Fy(q)1307 3842 y Fq(r)1344 3828 y Ft(\()p Fy(t)p Ft(\))31 b(of)g Fy(q)1623 3842 y Fq(r)1660 3828 y Ft(\()p Fy(x)p Ft(;)15 b Fy(t)p Ft(\))31 b(b)m(y)1525 4031 y(~)-52 b Fy(q)1559 4045 y Fq(r)1597 4031 y Ft(\()p Fy(t)p Ft(\))25 b(=)1891 3970 y Fy(t)1924 3937 y Fq(r)p 1831 4010 190 4 v 1831 4094 a Fy(t)20 b Fm(\000)g Ft(1)2031 4031 y Fy(q)2072 4045 y Fq(r)2109 4031 y Ft(\()p Fy(t)2177 3994 y Fi(\000)p Fo(1)2272 4031 y Ft(\))75 4303 y(and)30 b(let)37 b(~)-52 b Fy(q)424 4318 y Fq(\025)469 4303 y Ft(\()p Fy(t)p Ft(\))26 b(=)31 b(~)-51 b Fy(q)735 4318 y Fq(\025)776 4327 y Ff(1)814 4303 y Ft(\()p Fy(t)p Ft(\))7 b(~)-52 b Fy(q)958 4318 y Fq(\025)999 4327 y Ff(1)1038 4303 y Ft(\()p Fy(t)p Ft(\))15 b Fm(\001)g(\001)g(\001)24 b Ft(~)-52 b Fy(q)1319 4318 y Fq(\025)1360 4330 y Fu(`)1394 4303 y Ft(\()p Fy(t)p Ft(\))26 b(=)1765 4241 y Fy(t)1798 4208 y Fi(j)p Fq(\026)p Fi(j)p 1629 4282 391 4 v 1629 4369 a Ft(\()p Fy(t)20 b Fm(\000)g Ft(1\))1888 4342 y Fq(`)p Fo(\()p Fq(\026)p Fo(\))2029 4303 y Fy(q)2070 4317 y Fq(\026)2116 4303 y Ft(\()p Fy(t)2184 4270 y Fi(\000)p Fo(1)2279 4303 y Ft(\).)75 4431 y SDict begin H.S end 75 4431 a 75 4431 a SDict begin 13 H.A end 75 4431 a 75 4431 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.1) cvn /DEST pdfmark end 75 4431 a 0 TeXcolorgray 108 x Fz(Prop)s(osition)36 b(2.1.)p 0 TeXcolorgray 42 w Ft([)p 0 1 0 0 TeXcolorcmyk 869 4539 a SDict begin H.S end 869 4539 a 0 1 0 0 TeXcolorcmyk Ft(R91)p 0 1 0 0 TeXcolorcmyk 1027 4477 a SDict begin H.R end 1027 4477 a 1027 4539 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.Ram) cvn H.B /ANN pdfmark end 1027 4539 a 0 TeXcolorgray Ft(])d Fn(The)g(irr)-5 b(e)g(ducible)33 b(char)-5 b(acter)35 b(of)e Fy(\037)2276 4506 y Fq(\025)2354 4539 y Fn(of)g Fy(H)2537 4553 y Fq(n)2583 4539 y Ft(\()p Fy(q)s Ft(\))g Fn(is)g(determine)-5 b(d)34 b(by)1519 4715 y Ft(~)-52 b Fy(q)1553 4729 y Fq(\026)1599 4715 y Ft(\()p Fy(q)s Ft(\))26 b(=)1835 4628 y Fl(X)1837 4826 y Fq(\025)p Fi(`)p Fq(n)1982 4715 y Fy(\037)2039 4677 y Fq(\025)2039 4737 y(\026)2085 4715 y Ft(\()p Fy(q)s Ft(\))p Fy(s)2242 4730 y Fq(\025)2288 4715 y Fy(:)3563 4715 y SDict begin H.S end 3563 4715 a 3563 4715 a SDict begin 16.25 H.A end 3563 4715 a 3563 4715 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.8) cvn /DEST pdfmark end 3563 4715 a Ft(\(2.8\))175 4972 y(Thanks)j(to)i(\()p 0 0 1 TeXcolorrgb 647 4972 a SDict begin H.S end 647 4972 a 0 0 1 TeXcolorrgb Ft(2.4)p 0 0 1 TeXcolorrgb 764 4913 a SDict begin H.R end 764 4913 a 764 4972 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end 764 4972 a 0 TeXcolorgray Ft(\),)f(w)m(e)h(conclude)f(that)904 5175 y Fy(\037)961 5137 y Fq(\025)961 5197 y(\026)1007 5175 y Ft(\()p Fy(q)s Ft(\))c(=)f Fm(h)7 b Ft(~)-52 b Fy(q)1319 5189 y Fq(\026)1365 5175 y Ft(\()p Fy(q)s Ft(\))p Fy(;)15 b(s)1562 5190 y Fq(\025)1608 5175 y Fm(i)26 b Ft(=)1930 5113 y Fy(q)1974 5080 y Fq(n)p 1775 5154 402 4 v 1775 5241 a Ft(\()p Fy(q)d Fm(\000)d Ft(1\))2045 5214 y Fq(`)p Fo(\()p Fq(\026)p Fo(\))2186 5175 y Fm(h)p Fy(q)2262 5189 y Fq(\026)2309 5175 y Ft(\()p Fy(x)p Ft(;)15 b Fy(q)2480 5137 y Fi(\000)p Fo(1)2575 5175 y Ft(\))p Fy(;)g(s)2693 5190 y Fq(\025)2738 5175 y Ft(\()p Fy(x)p Ft(\))p Fm(i)p Fy(:)3563 5175 y SDict begin H.S end 3563 5175 a 3563 5175 a SDict begin 16.25 H.A end 3563 5175 a 3563 5175 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.9) cvn /DEST pdfmark end 3563 5175 a Ft(\(2.9\))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 0 TeXcolorgray 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end 67 -8 a 0 TeXcolorgray 0 TeXcolorgray 75 66 a Fo(4)1106 b(NAIHUAN)23 b(JING,)h(YU)g(WU,)f(AND)g(NING)h(LIU)p 0 TeXcolorgray 75 183 a SDict begin H.S end 75 183 a 75 183 a SDict begin 13 H.A end 75 183 a 75 183 a SDict begin [/View [/XYZ H.V]/Dest (subsection.2.3) cvn /DEST pdfmark end 75 183 a 83 x Ft(2.3.)47 b Fz(V)-9 b(ertex)36 b(op)s(erator)g (realization)g(of)g(Sc)m(h)m(ur)g(functions.)47 b Ft(Let)31 b Fr(h)h Ft(b)s(e)e(the)i(in\014nite)d(dimensional)75 374 y(Heisen)m(b)s(erg)h(algebra)g(o)m(v)m(er)h Fh(C)54 b Ft(with)29 b(generators)i Fy(h)1845 388 y Fq(n)1893 374 y Ft(,)f Fy(n)25 b Fm(2)g Fh(Z)15 b Fm(n)20 b(f)p Ft(0)p Fm(g)32 b Ft(and)d(the)i(cen)m(tral)f(elemen)m(t)h Fy(c)f Ft(sub)5 b(ject)75 482 y(to)31 b(the)g(relation:)1480 637 y([)p Fy(h)1557 651 y Fq(m)1624 637 y Fy(;)15 b(h)1716 651 y Fq(n)1764 637 y Ft(])26 b(=)f Fy(\016)1951 651 y Fq(m;)p Fi(\000)p Fq(n)2135 637 y Fy(m)20 b Fm(\001)g Fy(c:)3518 637 y SDict begin H.S end 3518 637 a 3518 637 a SDict begin 16.25 H.A end 3518 637 a 3518 637 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.10) cvn /DEST pdfmark end 3518 637 a Ft(\(2.10\))175 802 y(As)34 b(is)g(w)m(ell)f(kno)m (wn,)h(there)h(is)f(a)g(basic)g(represen)m(tation)h(realized)f(on)g (the)h(space)g Fy(V)52 b Ft(=)32 b Fy(S)5 b(y)s(m)p Ft(\()p Fr(h)3512 769 y Fi(\000)3571 802 y Ft(\))35 b(for)75 910 y(the)27 b(algebra)g Fr(h)p Ft(,)g(where)f Fy(V)47 b Ft(denotes)27 b(the)g(symmetric)f(algebra)h(generated)h(b)m(y)e(the)h (elemen)m(ts)g Fy(h)3339 924 y Fi(\000)p Fq(n)3442 910 y Ft(,)f Fy(n)f Fm(2)g Fh(N)7 b Ft(.)75 1018 y(Explicitly)-8 b(,)32 b(for)i(an)m(y)g Fy(n)d Fm(2)g Fh(N)7 b Ft(,)40 b(the)34 b(elemen)m(t)h Fy(h)1685 1032 y Fi(\000)p Fq(n)1821 1018 y Ft(acts)g(as)g(the)f(m)m(ultiplication)d(b)m(y)j Fy(h)3057 1032 y Fi(\000)p Fq(n)3159 1018 y Ft(,)g(while)f Fy(h)3512 1032 y Fq(n)3593 1018 y Ft(acts)75 1129 y(as)e(the)f (di\013eren)m(tiation)f(op)s(erator)i Fy(n)1426 1093 y Fq(@)p 1361 1108 173 4 v 1361 1160 a(@)t(h)1443 1169 y Fe(\000)p Fu(n)1543 1129 y Ft(,)f(whic)m(h)f(then)h(satisfy)g(the)h (relation)e(\()p 0 0 1 TeXcolorrgb 2874 1129 a SDict begin H.S end 2874 1129 a 0 0 1 TeXcolorrgb Ft(2.10)p 0 0 1 TeXcolorrgb 3036 1070 a SDict begin H.R end 3036 1070 a 3036 1129 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 3036 1129 a 0 TeXcolorgray Ft(\))i(with)e Fy(c)d Ft(=)f(1.)175 1248 y(De\014ne)30 b(the)h(canonical)f(inner)e(pro)s(duct)h(on)i Fy(V)50 b Ft(b)m(y)1492 1403 y Fm(h)p Fy(h)1579 1418 y Fi(\000)p Fq(\025)1680 1403 y Fy(;)15 b(h)1772 1417 y Fi(\000)p Fq(\026)1874 1403 y Fm(i)1909 1417 y Fq(V)1996 1403 y Ft(=)25 b Fy(\016)2132 1418 y Fq(\025\026)2220 1403 y Fy(z)2262 1418 y Fq(\025)2308 1403 y Fy(;)3518 1403 y SDict begin H.S end 3518 1403 a 3518 1403 a SDict begin 16.25 H.A end 3518 1403 a 3518 1403 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11) cvn /DEST pdfmark end 3518 1403 a Ft(\(2.11\))75 1558 y(where)30 b Fy(h)390 1573 y Fi(\000)p Fq(\025)516 1558 y Ft(:=)25 b Fy(h)689 1573 y Fi(\000)p Fq(\025)785 1582 y Ff(1)824 1558 y Fy(h)876 1573 y Fi(\000)p Fq(\025)972 1582 y Ff(2)1026 1558 y Fm(\001)15 b(\001)g(\001)175 1666 y Ft(Therefore,)30 b Fy(h)664 1680 y Fi(\000)p Fq(n)797 1666 y Ft(and)f Fy(h)1025 1680 y Fq(n)1103 1666 y Ft(are)i(dual)e(to)i(eac)m(h)g(other) g(with)e(resp)s(ect)h(to)i(\()p 0 0 1 TeXcolorrgb 2675 1666 a SDict begin H.S end 2675 1666 a 0 0 1 TeXcolorrgb Ft(2.11)p 0 0 1 TeXcolorrgb 2837 1608 a SDict begin H.R end 2837 1608 a 2837 1666 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.11) cvn H.B /ANN pdfmark end 2837 1666 a 0 TeXcolorgray Ft(\).)e(Namely)-8 b(,)1083 1822 y Fm(h)p Fy(h)1170 1836 y Fi(\000)p Fq(n)1273 1822 y Fy(f)5 b(;)15 b(g)s Fm(i)1444 1836 y Fq(V)1531 1822 y Ft(=)25 b Fm(h)p Fy(f)5 b(;)15 b(h)1804 1836 y Fq(n)1851 1822 y Fy(g)s Fm(i)1932 1836 y Fq(V)2085 1822 y Ft(for)30 b(an)m(y)h Fy(f)5 b(;)15 b(g)28 b Fm(2)d Fy(V)20 b(:)3518 1822 y SDict begin H.S end 3518 1822 a 3518 1822 a SDict begin 16.25 H.A end 3518 1822 a 3518 1822 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.12) cvn /DEST pdfmark end 3518 1822 a Ft(\(2.12\))175 1990 y(De\014ne)30 b(the)h(v)m(ertex)g (op)s(erators)g Fy(X)1366 2004 y Fq(n)1443 1990 y Ft(and)f Fy(X)1702 1957 y Fi(\003)1695 2012 y Fq(n)1742 1990 y Ft(:)h Fy(V)45 b Fm(\000)-15 b(!)25 b Fy(V)20 b Ft([[)p Fy(z)t(;)15 b(z)2323 1957 y Fi(\000)p Fo(1)2419 1990 y Ft(]])31 b(b)m(y)655 2255 y Fy(X)7 b Ft(\()p Fy(z)t Ft(\))27 b(=)e(exp)1130 2072 y Fl(0)1130 2236 y(@)1210 2168 y(X)1209 2364 y Fq(n)p Fi(\025)p Fo(1)1372 2193 y Ft(1)p 1367 2234 55 4 v 1367 2317 a Fy(n)1432 2255 y(h)1484 2269 y Fi(\000)p Fq(n)1586 2255 y Fy(z)1632 2217 y Fq(n)1679 2072 y Fl(1)1679 2236 y(A)1774 2255 y Ft(exp)1928 2072 y Fl(0)1928 2236 y(@)2008 2255 y Fm(\000)2094 2168 y Fl(X)2094 2364 y Fq(n)p Fi(\025)p Fo(1)2256 2193 y Ft(1)p 2252 2234 V 2252 2317 a Fy(n)2316 2255 y(h)2368 2269 y Fq(n)2416 2255 y Fy(z)2462 2217 y Fi(\000)p Fq(n)2564 2072 y Fl(1)2564 2236 y(A)2668 2255 y Ft(=)2766 2168 y Fl(X)2764 2366 y Fq(n)p Fi(2)p Fg(Z)2914 2255 y Fy(X)2989 2269 y Fq(n)3036 2255 y Fy(z)3082 2217 y Fq(n)3129 2255 y Fy(;)3518 2255 y SDict begin H.S end 3518 2255 a 3518 2255 a SDict begin 16.25 H.A end 3518 2255 a 3518 2255 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.13) cvn /DEST pdfmark end 3518 2255 a Ft(\(2.13\))616 2617 y Fy(X)698 2580 y Fi(\003)738 2617 y Ft(\()p Fy(z)t Ft(\))h(=)f(exp)1130 2435 y Fl(0)1130 2598 y(@)1209 2617 y Fm(\000)1296 2531 y Fl(X)1295 2727 y Fq(n)p Fi(\025)p Fo(1)1458 2556 y Ft(1)p 1453 2596 55 4 v 1453 2680 a Fy(n)1518 2617 y(h)1570 2631 y Fi(\000)p Fq(n)1672 2617 y Fy(z)1718 2580 y Fq(n)1765 2435 y Fl(1)1765 2598 y(A)1860 2617 y Ft(exp)2014 2435 y Fl(0)2014 2598 y(@)2094 2531 y(X)2094 2727 y Fq(n)p Fi(\025)p Fo(1)2256 2556 y Ft(1)p 2252 2596 V 2252 2680 a Fy(n)2316 2617 y(h)2368 2631 y Fq(n)2416 2617 y Fy(z)2462 2580 y Fi(\000)p Fq(n)2564 2435 y Fl(1)2564 2598 y(A)2668 2617 y Ft(=)2766 2531 y Fl(X)2764 2728 y Fq(n)p Fi(2)p Fg(Z)2914 2617 y Fy(X)2996 2580 y Fi(\003)2989 2640 y Fq(n)3036 2617 y Fy(z)3082 2580 y Fi(\000)p Fq(n)3184 2617 y Fy(:)3518 2617 y SDict begin H.S end 3518 2617 a 3518 2617 a SDict begin 16.25 H.A end 3518 2617 a 3518 2617 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.14) cvn /DEST pdfmark end 3518 2617 a Ft(\(2.14\))75 2877 y(Recall)30 b(that)h(\003)609 2891 y Fg(C)692 2877 y Ft(denotes)g(the)g(space)g(of) g(symmetric)f(functions.)g(The)g Fn(char)-5 b(acteristic)35 b(mapping)d Fy(\023)e Ft(from)75 2985 y Fy(V)51 b Ft(to)31 b(\003)353 2999 y Fg(C)435 2985 y Ft(is)f(de\014ned)f(b)m(y)984 3140 y Fy(\023)p Ft(\()p Fy(h)1103 3155 y Fi(\000)p Fq(\025)1204 3140 y Ft(\))d(:=)f Fy(p)1432 3155 y Fq(\025)1473 3164 y Ff(1)1527 3140 y Fm(\001)15 b(\001)g(\001)h Fy(p)1694 3155 y Fq(\025)1735 3167 y Fu(`)1795 3140 y Ft(=)25 b Fy(p)1937 3155 y Fq(\025)2072 3140 y Ft(for)31 b(an)m(y)f(partition)f Fy(\025:)3518 3140 y SDict begin H.S end 3518 3140 a 3518 3140 a SDict begin 16.25 H.A end 3518 3140 a 3518 3140 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.15) cvn /DEST pdfmark end 3518 3140 a Ft(\(2.15\))75 3315 y(By)k(de\014nition,)d Fy(\023)p Ft(\()p Fy(h)770 3329 y Fq(n)818 3315 y Ft(\))f(=)f Fy(\023)p Ft(\()p Fy(n)1179 3279 y Fq(@)p 1113 3294 173 4 v 1113 3346 a(@)t(h)1195 3355 y Fe(\000)p Fu(n)1296 3315 y Ft(\))h(=)f Fy(n)1563 3279 y Fq(@)p 1524 3294 120 4 v 1524 3346 a(@)t(p)1601 3354 y Fu(n)1681 3315 y Ft(:=)h Fy(p)1852 3282 y Fi(\003)1852 3337 y Fq(n)1899 3315 y Ft(.)j(It)h(is)e(sho)m(w)m(ed)i([)p 0 1 0 0 TeXcolorcmyk 2491 3315 a SDict begin H.S end 2491 3315 a 0 1 0 0 TeXcolorcmyk Ft(J91b)p 0 1 0 0 TeXcolorcmyk 2679 3251 a SDict begin H.R end 2679 3251 a 2679 3315 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.J2) cvn H.B /ANN pdfmark end 2679 3315 a 0 TeXcolorgray Ft(,)g(Theorem)f(3.6])i(that)f Fy(\023)g Ft(is)e(an)75 3433 y(isometric)f(isomorphism)d(b)s(et)m(w)m (een)k Fy(V)51 b Ft(and)30 b(\003)1691 3447 y Fg(C)1743 3433 y Ft(.)175 3541 y(It)g(follo)m(ws)g(from)f(\()p 0 0 1 TeXcolorrgb 821 3541 a SDict begin H.S end 821 3541 a 0 0 1 TeXcolorrgb Ft(2.12)p 0 0 1 TeXcolorrgb 983 3483 a SDict begin H.R end 983 3483 a 983 3541 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 983 3541 a 0 TeXcolorgray Ft(\))i(that)1156 3697 y Fm(h)p Fy(p)1237 3711 y Fq(n)1284 3697 y Fy(f)5 b(;)15 b(g)s Fm(i)25 b Ft(=)g Fm(h)p Fy(f)5 b(;)15 b(p)1747 3659 y Fi(\003)1747 3719 y Fq(n)1794 3697 y Fy(g)s Fm(i)92 b Ft(for)30 b(an)m(y)h Fy(f)5 b(;)15 b(g)28 b Fm(2)d Ft(\003)2588 3711 y Fg(Q)2644 3697 y Fy(:)3518 3697 y SDict begin H.S end 3518 3697 a 3518 3697 a SDict begin 16.25 H.A end 3518 3697 a 3518 3697 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.16) cvn /DEST pdfmark end 3518 3697 a Ft(\(2.16\))75 3852 y(Note)32 b(that)f Fm(\003)f Ft(is)g Fh(C)17 b Ft(-)q(linear)35 b(and)29 b(an)m(ti-in)m(v)m(olutiv)m (e.)175 3960 y(Under)36 b(the)i(c)m(haracteristic)g(mapping,)e(the)h (images)h(of)f Fy(X)7 b Ft(\()p Fy(z)t Ft(\))39 b(and)e Fy(X)2675 3927 y Fi(\003)2715 3960 y Ft(\()p Fy(z)t Ft(\))h(\(denoted)g (b)m(y)f Fy(S)5 b Ft(\()p Fy(z)t Ft(\))39 b(and)75 4068 y Fy(S)136 4035 y Fi(\003)175 4068 y Ft(\()p Fy(z)t Ft(\))32 b(resp)s(ectiv)m(ely\))e(are)h(giv)m(en)f(b)m(y)402 4333 y Fy(\023)p Ft(\()p Fy(X)7 b Ft(\()p Fy(z)t Ft(\)\))28 b(=)d Fy(S)5 b Ft(\()p Fy(z)t Ft(\))26 b(=)f(exp)1279 4150 y Fl(0)1279 4314 y(@)1359 4246 y(X)1358 4442 y Fq(n)p Fi(\025)p Fo(1)1521 4271 y Ft(1)p 1516 4312 55 4 v 1516 4395 a Fy(n)1581 4333 y(p)1627 4347 y Fq(n)1674 4333 y Fy(z)1720 4295 y Fq(n)1767 4150 y Fl(1)1767 4314 y(A)1862 4333 y Ft(exp)2016 4150 y Fl(0)2016 4314 y(@)2095 4333 y Fm(\000)2182 4246 y Fl(X)2181 4442 y Fq(n)p Fi(\025)p Fo(1)2386 4271 y Fy(@)p 2339 4312 147 4 v 2339 4395 a(@)5 b(p)2438 4409 y Fq(n)2495 4333 y Fy(z)2541 4295 y Fi(\000)p Fq(n)2643 4150 y Fl(1)2643 4314 y(A)2748 4333 y Ft(=)2845 4246 y Fl(X)2844 4444 y Fq(n)p Fi(2)p Fg(Z)2993 4333 y Fy(S)3049 4347 y Fq(n)3096 4333 y Fy(z)3142 4295 y Fq(n)3189 4333 y Fy(;)3518 4333 y SDict begin H.S end 3518 4333 a 3518 4333 a SDict begin 16.25 H.A end 3518 4333 a 3518 4333 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.17) cvn /DEST pdfmark end 3518 4333 a Ft(\(2.17\))323 4695 y Fy(\023)p Ft(\()p Fy(X)472 4658 y Fi(\003)513 4695 y Ft(\()p Fy(z)t Ft(\)\))26 b(=)f Fy(S)847 4658 y Fi(\003)887 4695 y Ft(\()p Fy(z)t Ft(\))h(=)f(exp)1279 4512 y Fl(0)1279 4676 y(@)1358 4695 y Fm(\000)1445 4609 y Fl(X)1444 4805 y Fq(n)p Fi(\025)p Fo(1)1607 4634 y Ft(1)p 1602 4674 55 4 v 1602 4758 a Fy(n)1667 4695 y(p)1713 4709 y Fq(n)1760 4695 y Fy(z)1806 4658 y Fq(n)1853 4512 y Fl(1)1853 4676 y(A)1948 4695 y Ft(exp)2102 4512 y Fl(0)2102 4676 y(@)2182 4609 y(X)2181 4805 y Fq(n)p Fi(\025)p Fo(1)2386 4634 y Fy(@)p 2339 4674 147 4 v 2339 4758 a(@)5 b(p)2438 4772 y Fq(n)2495 4695 y Fy(z)2541 4658 y Fi(\000)p Fq(n)2643 4512 y Fl(1)2643 4676 y(A)2748 4695 y Ft(=)2845 4609 y Fl(X)2844 4806 y Fq(n)p Fi(2)p Fg(Z)2993 4695 y Fy(S)3054 4658 y Fi(\003)3049 4718 y Fq(n)3096 4695 y Fy(z)3142 4658 y Fi(\000)p Fq(n)3244 4695 y Fy(:)3518 4695 y SDict begin H.S end 3518 4695 a 3518 4695 a SDict begin 16.25 H.A end 3518 4695 a 3518 4695 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.18) cvn /DEST pdfmark end 3518 4695 a Ft(\(2.18\))175 4955 y(Let)33 b(us)g(recall)g(the)g(v)m(ertex)i(op)s (erator)e(realization)g(of)g(Sc)m(h)m(ur)g(functions)f(and)g(the)i (relations)e(b)s(et)m(w)m(een)75 5063 y Fy(S)131 5077 y Fq(n)208 5063 y Ft(and)e Fy(S)446 5030 y Fi(\003)441 5085 y Fq(m)507 5063 y Ft(.)75 5122 y SDict begin H.S end 75 5122 a 75 5122 a SDict begin 13 H.A end 75 5122 a 75 5122 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.2) cvn /DEST pdfmark end 75 5122 a 0 TeXcolorgray 108 x Fz(Prop)s(osition)36 b(2.2.)p 0 TeXcolorgray 42 w Ft([)p 0 1 0 0 TeXcolorcmyk 869 5230 a SDict begin H.S end 869 5230 a 0 1 0 0 TeXcolorcmyk Ft(J91a)p 0 1 0 0 TeXcolorcmyk 1053 5168 a SDict begin H.R end 1053 5168 a 1053 5230 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.J1) cvn H.B /ANN pdfmark end 1053 5230 a 0 TeXcolorgray Ft(])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 0 TeXcolorgray 67 -8 a SDict begin H.S end 67 -8 a 67 -8 a SDict begin H.R end 67 -8 a 67 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end 67 -8 a 0 TeXcolorgray 0 TeXcolorgray 558 66 a Fo(DUAL)30 b(MURNA)n(GHAN-NAKA)-6 b(Y)g(AMA)28 b(R)n(ULE)j(F)n(OR)f (HECKE)g(ALGEBRAS)h(IN)g(TYPE)f Fq(A)448 b Fo(5)p 0 TeXcolorgray 75 183 a SDict begin H.S end 75 183 a 75 183 a SDict begin 13 H.A end 75 183 a 75 183 a SDict begin [/View [/XYZ H.V]/Dest (Item.1) cvn /DEST pdfmark end 75 183 a 0 TeXcolorgray 215 266 a Fn(\(1\))p 0 TeXcolorgray 43 w(F)-7 b(or)47 b(any)g(c)-5 b(omp)g(osition)49 b Fy(\026)h Ft(=)f(\()p Fy(\026)1577 280 y Fo(1)1617 266 y Fy(;)15 b(:)g(:)g(:)i(;)e(\026)1874 281 y Fq(k)1916 266 y Ft(\))p Fn(,)47 b(the)f(pr)-5 b(o)g(duct)49 b Fy(S)2582 280 y Fq(\026)2628 266 y Fy(:)p Ft(1)i(:=)f Fy(S)2951 280 y Fq(\026)2993 289 y Ff(1)3047 266 y Fm(\001)15 b(\001)g(\001)h Fy(S)3224 280 y Fq(\026)3266 292 y Fu(k)3308 266 y Fy(:)p Ft(1)51 b(=)f Fy(s)3593 280 y Fq(\026)3685 266 y Fn(is)378 374 y(the)42 b(Schur)g(function)g(lab)-5 b(ele)g(d)43 b(by)f Fy(\026)p Fn(.)f(In)h(gener)-5 b(al,)42 b Fy(s)2242 388 y Fq(\026)2330 374 y Ft(=)g(0)g Fn(or)g Fm(\006)p Fy(s)2770 389 y Fq(\025)2857 374 y Fn(for)g(a)g(p)-5 b(artition)44 b 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Ft(\(2.20\))1384 1131 y Fy(S)1440 1145 y Fq(m)1506 1131 y Fy(S)1567 1093 y Fi(\003)1562 1153 y Fq(n)1629 1131 y Ft(+)20 b Fy(S)1781 1093 y Fi(\003)1776 1153 y Fq(n)p Fi(\000)p Fo(1)1913 1131 y Fy(S)1969 1145 y Fq(m)p Fi(\000)p Fo(1)2151 1131 y Ft(=)25 b Fy(\016)2287 1145 y Fq(m;n)2416 1131 y Fy(:)3518 1131 y SDict begin H.S end 3518 1131 a 3518 1131 a SDict begin 16.25 H.A end 3518 1131 a 3518 1131 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.21) cvn /DEST pdfmark end 3518 1131 a Ft(\(2.21\))75 1194 y SDict begin H.S end 75 1194 a 75 1194 a SDict begin 13 H.A end 75 1194 a 75 1194 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.3) cvn /DEST pdfmark end 75 1194 a 0 TeXcolorgray 108 x Fz(Remark)i(2.3.)p 0 TeXcolorgray 37 w Fn(Supp)-5 b(ose)28 b Fy(k)g Fm(2)d Fh(Z)p Fn(.)d(Then)27 b Fy(S)1572 1269 y Fi(\003)1567 1329 y(\000)p Fq(k)1664 1302 y Fy(:)p Ft(1)g Fn(is)g(exactly)g(the)g(elementary)h(symmetric)g (function)e Fy(e)3584 1317 y Fq(k)3627 1302 y Ft(\()p Fy(x)p Ft(\))75 1413 y 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