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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="representation_theory">Representation theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li><a class="existingWikiWord" 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><a class="existingWikiWord" href="/nlab/show/Beilinson-Bernstein+localization">Beilinson-Bernstein localization</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/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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#historical_idea_with_pedagogic_overtones'>Historical Idea (with pedagogic overtones)</a></li> <ul> <li><a href='#representing_groups'>Representing Groups</a></li> </ul> <li><a href='#general_definition'>General definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of <em>representation</em> is closely related to, or even identical, to that of <a class="existingWikiWord" href="/nlab/show/action">action</a>: some object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> that has a notion of <a class="existingWikiWord" href="/nlab/show/composition">composition</a> is <em>represented</em> on some object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> that has a notion of composition. In this generality <em>representation</em> is just another word for <a class="existingWikiWord" href="/nlab/show/functor">functor</a> (or potentially <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/infinity-functor">functor</a>). But in practice the term <em>representation</em> is typically used in the context of <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, where we attempt to study <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in terms of its representations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is typically rather more familiar.</p> <p>Most specifically, one studies representations of a <a class="existingWikiWord" href="/nlab/show/group">group</a> by linear endomorphisms of a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is (the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of) a group and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>. However, the typical tools in representation theory these days involve vast generalizations of the notion of a linear representation of a group; for instance, one studies <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>s on <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>Ad</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">G //_{Ad}G</annotation></semantics></math> and things like that. This may be thought of as studying representations with values in <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-vector+bundle">∞-vector space</a>s.</p> <h2 id="historical_idea_with_pedagogic_overtones">Historical Idea (with pedagogic overtones)</h2> <h3 id="representing_groups">Representing Groups</h3> <p>The notion of a <a class="existingWikiWord" href="/nlab/show/group">group</a> of ‘operators’ was already being used in about 1832 in work by Galois, and others, but there was not a definition of an abstract group until 40 years later when Cayley wrote:</p> <blockquote> <p><em>A group is defined by the law of composition of its members.</em></p> </blockquote> <p>(see the article on <a href="http://www-history.mcs.st-and.ac.uk/HistTopics/Abstract_groups.html">The abstract group concept</a> in the St Andrews History of Mathematics archive.) Groups as well behaved sets of functions were beginning to be well understood and used, for instance in Klein’s work on geometry. Cayley proved that every finite group could be realised as a group of permutations. The theory of representations grew from that.</p> <p>An abstract group could be studied by mapping it into a group of permutations or of invertible matrices, as then you could bring techniques from one area of mathematics (linear algebra) to the assistance of another, the Theory of Abstract Groups. This was exploited fully by Frobenius, Burnside, Schur and later Bauer.</p> <p>That is one theme: you take an abstract algebraic thing and study it by mapping it into a similar structure which you think you know more about!</p> <p>This also uses another basic idea from the start of group theory. The earlier pioneers thought of groups as groups of ‘operators’, but that means they have to operate on something. To make things more explicit, let <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> be a finite group then we know there are homomorphisms from <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> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math>. (We could take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> to be the order of the group and use Cayley’s theorem but we do not assume that is the case.) If we look at such a homomorphism we get an <a class="existingWikiWord" href="/nlab/show/action">action</a> 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 an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-element set.</p> <p>Similarly if we take a homomorphism from <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> into a group of invertible matrices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Gl</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gl_n(K)</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> a field, say, then we get a linear action 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 the vector space, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">K^n</annotation></semantics></math>.</p> <p>As you would expect, we can generalise and categorify this basic idea in several useful ways.</p> <p>We can think 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> as a groupoid, <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>, (the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> 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>), and then a linear representation / action will be a functor from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math>, the category of vector spaces over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. We could replace <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> by a general groupoid, or a general category, but then a representation of that is the same as a diagram of that ‘shape’ in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math>. We could replace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math> by another more general category, or higher category, but if we are thinking of diagrams as representations, perhaps we should not totally forget that the term ‘representation’ did mean a process whereby the perhaps abstract ‘syntactical’ objects of the category gain a ‘semantic’ meaning, as ‘operations’ of some type, and which in turn, can be usefully used to gain information on the inherent structure.</p> <h2 id="general_definition">General definition</h2> <p>In a rather general form, we therefore have a <strong>representation</strong> of a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is simply a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F\colon C \to D</annotation></semantics></math>. Similarly, an <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> between representations (“<a class="existingWikiWord" href="/nlab/show/intertwiner">intertwiner</a>”) is simply a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> between functors when they are being thought of as representations. Thus we have a <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a>.</p> <p>The term ‘representation’ is most often used when one or more of the following conditions apply:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of vector spaces over some <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>; one then has a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-linear representation</strong>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of a <a class="existingWikiWord" href="/nlab/show/group">group</a>; one then has a <strong>group representation</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>. Such a representation gives us a specific object <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>; we say that we have a representation 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 <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>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+category">free category</a> on a <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a>; one then has a <strong><a class="existingWikiWord" href="/nlab/show/quiver+representation">quiver representation</a></strong>.</li> </ul> <p>The classical <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> of groups is about representations of (finite, topological, smooth etc.) groups on (topological) vector spaces, that is when the first two conditions apply.</p> <p>There are also enriched, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-linear and other versions, hence one can talk about representations of <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebras">vertex operator algebras</a>, etc. See also <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+representation">trivial representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a>, <a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+representation">regular representation</a>, <a class="existingWikiWord" href="/nlab/show/alternating+representation">alternating representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+representation">adjoint representation</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+representation">coadjoint representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a>, <a class="existingWikiWord" href="/nlab/show/coinduced+representation">coinduced representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+representation">Lie algebra representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+representation">Galois representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+representation">groupoid representation</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quiver+variety">quiver variety</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/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><strong>representation</strong>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+representation">faithful representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+representation">virtual representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-representation">star-representation</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/representation+of+a+C-star-algebra">representation of a C-star-algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+sphere">representation sphere</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/character">character</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/singleton+representation">singleton representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/global+equivariant+stable+homotopy+theory">global equivariant stable homotopy theory</a></p> </li> </ul> <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="references">References</h2> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on May 19, 2022 at 15:05:17. See the <a href="/nlab/history/representation" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/representation" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/7245/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/representation/36" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/representation" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/representation" accesskey="S" class="navlink" id="history" rel="nofollow">History (36 revisions)</a> <a href="/nlab/show/representation/cite" style="color: black">Cite</a> <a href="/nlab/print/representation" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/representation" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>