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href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+object">equivariant object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/socle">socle</a>, <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+algebra">module algebra</a>, <a class="existingWikiWord" href="/nlab/show/comodule+algebra">comodule algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+action">Hopf action</a>, <a class="existingWikiWord" href="/nlab/show/measuring">measuring</a></p> </li> </ul> <h2 id="geometric_representation_theory">Geometric representation theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/perverse+sheaf">perverse sheaf</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, <a class="existingWikiWord" href="/nlab/show/lambda-ring">lambda-ring</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+function">symmetric function</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>, <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>, <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstruction+theorems">reconstruction theorems</a></p> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> </li> <li> <p><span class="newWikiWord">Be?linson-Bernstein localization<a href="/nlab/new/Be%3Flinson-Bernstein+localization">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/representation+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="mathematics">Mathematics</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/math+resources">math resources</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/history+of+mathematics">history of mathematics</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/foundations">Structural Foundations</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logic">logic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a></li> <li><a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a></li> <li><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a></li> <li><a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a></li> <li><a href="http://ncatlab.org/nlab/list/foundational+axiom">category:foundational axiom</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">(∞,1)-topos theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></li> <li><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a></li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topology+and+geometry">Topology and Geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> (general list), <a class="existingWikiWord" href="/nlab/show/topology">topology</a> (general list)</li> <li><a class="existingWikiWord" href="/nlab/show/general+topology">general topology</a></li> <li><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></li> <li><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a></li> <li><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a> (general flavour)</li> <li><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">Algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a>, <a class="existingWikiWord" href="/nlab/show/ring+theory">ring theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+approaches+to+differential+calculus">algebraic approaches to differential calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterexamples+in+algebra">counterexamples in algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analysis">analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/infinity-Lie+theory+-+contents">higher Lie theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+theory">probability theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+mathematics">discrete mathematics</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#InHomotopyTypeTheory'>In homotopy type theory</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Representation theory</em> is concerned with the study of <a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> <a class="existingWikiWord" href="/nlab/show/structures">structures</a> via their <a class="existingWikiWord" href="/nlab/show/representations">representations</a>. This concerns notably <a class="existingWikiWord" href="/nlab/show/groups">groups</a>, directly or in their incarnation as <a class="existingWikiWord" href="/nlab/show/group+algebras">group algebras</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebras">Hopf algebras</a> or <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a>, and usually concerns <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear</a> <a class="existingWikiWord" href="/nlab/show/representations">representations</a>, hence <a class="existingWikiWord" href="/nlab/show/modules">modules</a> of these structures. But more generally representation theory also studies <a class="existingWikiWord" href="/nlab/show/representations">representations</a>/<a class="existingWikiWord" href="/nlab/show/modules">modules</a>/<a class="existingWikiWord" href="/nlab/show/actions">actions</a> of generalizations of such structures, such as <a class="existingWikiWord" href="/nlab/show/coalgebras">coalgebras</a> via their <a class="existingWikiWord" href="/nlab/show/comodules">comodules</a> etc.</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a></em>.</p> <h2 id="InHomotopyTypeTheory">In homotopy type theory</h2> <p>The fundamental concepts of representation theory have a particular natural formulation in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> and in fact in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, which also refines it from the study of <a class="existingWikiWord" href="/nlab/show/representations">representations</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> to that of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representations">∞-representations</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a>. This includes both <a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groups">discrete ∞-groups</a> as well as <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+types">geometric homotopy types</a> such as <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groups">smooth ∞-groups</a>, the higher analog of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>.</p> <p>The key observation to this translation is that</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is equivalently given by its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> regarded with its canonical <a class="existingWikiWord" href="/nlab/show/pointed+object">point</a> (see at <a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a>), hence the universal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G &\longrightarrow& \ast \\ && \downarrow \\ && \mathbf{B}G } </annotation></semantics></math></div></li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on any <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+type">geometric homotopy type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is equivalently given by a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ρ</mi></msub><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ V &\stackrel{}{\longrightarrow}& V//_\rho G \\ && \downarrow \\ && \mathbf{B}G } \,, </annotation></semantics></math></div> <p>hence by a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> which is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a> to the universal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> (see at <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></em> for more on this).</p> </li> </ol> <p>Under this identification, the representation theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is equivalently</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> in the <a class="existingWikiWord" href="/nlab/show/context">context</a> of/<a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent on</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>.</p> </li> </ul> <p>More in detail, this yields the following identifications:</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> and <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> in terms of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong> (<a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">FSS 12 I, exmp. 4.4</a>):</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></th><th><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pointed+type">pointed</a> <a class="existingWikiWord" href="/nlab/show/connected+homotopy+type">connected</a> <a class="existingWikiWord" href="/nlab/show/context">context</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinvariants">coinvariants</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trivial+representation">trivial representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+invariants">homotopy invariants</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+cohomology">∞-group cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> of <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/restricted+representation">restricted representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinduced+representation">coinduced representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a> in <a class="existingWikiWord" href="/nlab/show/context">context</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum+with+G-action">spectrum with G-action</a> (<a class="existingWikiWord" href="/nlab/show/naive+G-spectrum">naive G-spectrum</a>)</td></tr> </tbody></table> </div> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory+of+the+symmetric+group">representation theory of the symmetric group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory+of+the+general+linear+group">representation theory of the general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory+of+the+special+unitary+group">representation theory of the special unitary group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schur-Weyl+duality">Schur-Weyl duality</a></p> </li> </ul> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/module">module</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/character">character</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+object">equivariant object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weight+%28in+representation+theory%29">weight (in representation theory)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/socle">socle</a>, <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+algebra">module algebra</a>, <a class="existingWikiWord" href="/nlab/show/comodule+algebra">comodule algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+action">Hopf action</a>, <a class="existingWikiWord" href="/nlab/show/measuring">measuring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/irreducible+representation">irreducible representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schur%27s+lemma">Schur's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Young+diagram">Young diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schur+index">Schur index</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/McKay+correspondence">McKay correspondence</a>, <a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE classification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/asymptotic+representation+theory">asymptotic representation theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil+theorem">Borel-Weil theorem</a>, <a class="existingWikiWord" href="/nlab/show/Beilinson-Bernstein+localization">Beilinson-Bernstein localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/perverse+sheaf">perverse sheaf</a>, <a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+induction">Dirac induction</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verma+module">Verma module</a>, <a class="existingWikiWord" href="/nlab/show/BGG+resolution">BGG resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+stability">representation stability</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, <a class="existingWikiWord" href="/nlab/show/lambda-ring">lambda-ring</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+function">symmetric function</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><span class="newWikiWord">character sheaf<a href="/nlab/new/character+sheaf">?</a></span>, <a class="existingWikiWord" href="/nlab/show/Harish+Chandra+transform">Harish Chandra transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>, <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>, <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstruction+theorems">reconstruction theorems</a></p> </li> </ul> <h2 id="References">References</h2> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, <a class="existingWikiWord" href="/nlab/show/Dmitry+Vaintrob">Dmitry Vaintrob</a>, Elena Yudovina:</p> <p><em>Introduction to representation theory</em>, Student Mathematical Library <strong>59</strong>, AMS (2011) [<a href="https://arxiv.org/abs/0901.0827">arXiv:0901.0827</a>, <a href="https://bookstore.ams.org/stml-59">ams:stml-59</a>]</p> </li> <li id="tomDieck09"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <em>Representation theory</em> (2009) [<a href="http://www.uni-math.gwdg.de/tammo/rep.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/tomDieckRepresentationTheory.pdf" title="pdf">pdf</a>]</p> </li> <li id="Teleman05"> <p><a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>Representation theory</em>, lecture notes 2005 (<a href="https://math.berkeley.edu/~teleman/math/RepThry.pdf">pdf</a>)</p> </li> <li> <p>Joel Robbin, <em>Real, Complex and Quaternionic representations</em>, 2006 (<a href="http://www.math.wisc.edu/~robbin/angelic/RCH-G.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Robbin08RCHRep.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+R.+Shafarevich">Igor R. Shafarevich</a>, <a class="existingWikiWord" href="/nlab/show/Alexey+O.+Remizov">Alexey O. Remizov</a>: §14 in: <em>Linear Algebra and Geometry</em> (2012) [<a href="https://doi.org/10.1007/978-3-642-30994-6">doi:10.1007/978-3-642-30994-6</a>, <a href="https://maa.org/press/maa-reviews/linear-algebra-and-geometry">MAA-review</a>]</p> </li> </ul> <p>Textbook accounts</p> <p>for <a class="existingWikiWord" href="/nlab/show/finite+groups">finite groups</a>:</p> <ul> <li> <p>Charles Curtis, Irving Reiner, <em>Representation theory of finite groups and associative algebras</em>, AMS 1962</p> </li> <li id="LuxPahlings10"> <p>Klaus Lux, Herbert Pahlings, <em>Representations of groups – A computational approach</em>, Cambridge University Press 2010 (<a href="http://www.math.rwth-aachen.de/~RepresentationsOfGroups/">author page</a>, <a href="http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521768078">publisher page</a>)</p> </li> <li> <p>Caroline Gruson, Vera Serganova, <em>From Finite Groups to Quivers via Algebras – A Journey Through Representation Theory</em>, Springer (2018) [<a href="https://doi.org/10.1007/978-3-319-98271-7">doi:10.1007/978-3-319-98271-7</a>]</p> </li> </ul> <p>and more generally for <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <a class="existingWikiWord" href="/nlab/show/Theodor+Br%C3%B6cker">Theodor Bröcker</a>, <em>Representations of compact Lie groups</em>, Springer (1985) [<a href="https://link.springer.com/book/10.1007/978-3-662-12918-0">doi:10.1007/978-3-662-12918-0</a>]</p> </li> <li id="FultonHarris91"> <p><a class="existingWikiWord" href="/nlab/show/William+Fulton">William Fulton</a>, <a class="existingWikiWord" href="/nlab/show/Joe+Harris">Joe Harris</a>, <em>Representation Theory: a First Course</em>, Springer, Berlin, 1991 (<a href="https://link.springer.com/book/10.1007/978-1-4612-0979-9">doi:10.1007/978-1-4612-0979-9</a>)</p> </li> </ul> <p>In the context of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Woit">Peter Woit</a>, <em>Quantum Theory, Groups and Representations: An Introduction</em>, Springer 2017 [<a href="https://doi.org/10.1007/978-3-319-64612-1">doi:10.1007/978-3-319-64612-1</a>, ISBN:978-3-319-64610-7]</li> </ul> <p>Discussion via <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a>/<a class="existingWikiWord" href="/nlab/show/Penrose+notation">Penrose notation</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jeffrey+Ellis+Mandula">Jeffrey Ellis Mandula</a>, <em>Diagrammatic techniques in group theory</em>, Southampton Univ. Phys. Dept. (1981) (<a href="https://cds.cern.ch/record/129911">cds:129911</a>, <a href="https://cds.cern.ch/record/129911/files/SHEP%2080-81-7.pdf">pdf</a>)</p> </li> <li id="Cvitanovic08"> <p><a class="existingWikiWord" href="/nlab/show/Predrag+Cvitanovi%C4%87">Predrag Cvitanović</a>, <em>Group Theory: Birdtracks, Lie’s, and Exceptional Groups</em>, Princeton University Press July 2008 (<a href="https://press.princeton.edu/books/paperback/9780691202983/group-theory">PUP</a>, <a href="http://birdtracks.eu/">birdtracks.eu</a>, <a href="http://www.birdtracks.eu/version9.0/GroupTheory.pdf">pdf</a>)</p> <blockquote> <p>(aimed at <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a> and <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>)</p> </blockquote> </li> </ul> <p>Further references:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mikhail+Khovanov">Mikhail Khovanov</a>, <em><a href="http://www.math.columbia.edu/~khovanov/resources">Resources</a></em>.</li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/number+theory">number theory</a> and the <a class="existingWikiWord" href="/nlab/show/Langlands+program">Langlands program</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Robert+Langlands">Robert Langlands</a>, <em>Representation theory: Its rise and its role in number theory</em> (<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.207.3303">web</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 10, 2024 at 20:07:45. 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