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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#statement'>Statement</a></li> <li><a href='#proof'>Proof</a></li> <li><a href='#InterpretationInCategoricalAlgebra'>Interpretation in categorical algebra</a></li> <li><a href='#GeneralizationsAndVariants'>Generalizations and variants</a></li> <ul> <li><a href='#for_simple_modules'>For simple modules</a></li> <li><a href='#for_simple_objects_in_an_abelian_category'>For simple objects in an abelian category</a></li> <li><a href='#for_bridgeland_stable_objects'>For Bridgeland stable objects</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Schur’s lemma is one of the fundamental facts of <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>. It concerns basic properties of the <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> between <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible</a> <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a>.</p> <p>The lemma consists of two parts that depend on different assumptions (a distinction often not highlighted in the literature):</p> <ol> <li> <p>The first statement applies over <em>every</em> <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a>:</p> <p>It says that there are no non-<a class="existingWikiWord" href="/nlab/show/zero+morphism">zero</a> <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> between distinct (i.e. non-<a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a>) <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a> and any non-zero morphism among isomorphic irreducibles is an isomorphism.</p> </li> <li> <p>The second statement applies only in the special case that the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> is an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a> (such as the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>) and that the <a class="existingWikiWord" href="/nlab/show/representations">representations</a> are <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite-dimensional</a>:</p> <p>It says that, in this case, moreover the only non-trivial <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a> of an <a class="existingWikiWord" href="/nlab/show/irreducible+representation">irreducible representation</a> are multiples of the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a>.</p> </li> </ol> <h2 id="statement">Statement</h2> <p>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 <a class="existingWikiWord" href="/nlab/show/group">group</a>. In the following:</p> <ul> <li> <p><em>representation</em> means <em><a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</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></em>, linear over some <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a>,</p> </li> <li> <p><em>finite dimensional representation</em> means that the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> is a <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional vector space</a>,</p> </li> <li> <p>an <em><a class="existingWikiWord" href="/nlab/show/irreducible+representation">irreducible representation</a></em> is one whose only <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/invariant">invariant</a> <a class="existingWikiWord" href="/nlab/show/subspaces">subspaces</a> (<a class="existingWikiWord" href="/nlab/show/fixed+point+spaces">fixed point spaces</a>) are the trivial degenerate cases: the <a class="existingWikiWord" href="/nlab/show/zero+object">zero</a>-subspace and the full space itself.</p> </li> </ul> <p> <div class='num_prop' id='SchurLemma'> <h6>Proposition</h6> <p><strong>(Schur’s lemma)</strong> <br /></p> <ol> <li id="FirstItem"> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">\phi \;\colon\; V\to W</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a>, is either the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a> or an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p>It follows that the <a class="existingWikiWord" href="/nlab/show/endomorphism+ring">endomorphism ring</a> of an <a class="existingWikiWord" href="/nlab/show/irreducible+representation">irreducible representation</a> is a <a class="existingWikiWord" href="/nlab/show/division+ring">division ring</a>.</p> </li> <li id="SecondItem"> <p>In the case that the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> is an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a>, <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\phi \;\colon\; V \to V</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite dimensional</a> <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</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> are a multiples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>⋅</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">c \cdot id</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity operator</a>.</p> <p>In other words, nontrivial automorphisms of irreducible representations, <em>a priori</em> possible by <a href="#FirstItem">(1)</a>, are ruled out over algebraically closed fields.</p> </li> </ol> <p></p> </div> </p> <h2 id="proof">Proof</h2> <p>As it goes with very fundamental lemmas, the <a class="existingWikiWord" href="/nlab/show/proof">proof</a> of Schur’s lemma follows by elementary inspection.</p> <p> <div class='proof'> <h6>Proof</h6> <p>(of Prop. <a class="maruku-ref" href="#SchurLemma"></a>) <br /> For the first statement:</p> <p>It is immediate to see that both the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> as well as <a class="existingWikiWord" href="/nlab/show/image">image</a> of a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>W</mi></mrow><annotation encoding="application/x-tex"> V \overset{f}{\longrightarrow} W </annotation></semantics></math></div> <p>of any <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/representations">representations</a> are <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/invariant">invariant</a> <a class="existingWikiWord" href="/nlab/show/subspaces">subspaces</a> (<a class="existingWikiWord" href="/nlab/show/fixed+point+spaces">fixed point spaces</a>). But by the very definition of <a class="existingWikiWord" href="/nlab/show/irreducible+representation">irreducibility</a>, the only such subspaces of <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> are the degenerate ones: their zero subspaces and the full spaces themselves.</p> <p>Now if the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> is all of <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> or the <a class="existingWikiWord" href="/nlab/show/image">image</a> is <a class="existingWikiWord" href="/nlab/show/zero+object">zero</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a>. The only case left is that the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> is <a class="existingWikiWord" href="/nlab/show/zero+object">zero</a> <em>and</em> the <a class="existingWikiWord" href="/nlab/show/image">image</a> is all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>, but this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/injective+map">injective</a> and <a class="existingWikiWord" href="/nlab/show/surjective+map">surjective</a> and is hence an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p>For the second statement:</p> <p>Now we use that over an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed 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> of every <a class="existingWikiWord" href="/nlab/show/linear+map">linear</a> <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a> of a <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional vector space</a> has an <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">c \in k</annotation></semantics></math> (which is, ultimately, due to the <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+algebra">fundamental theorem of algebra</a> for <a class="existingWikiWord" href="/nlab/show/algebraically+closed+fields">algebraically closed fields</a>). Now, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> also the linear combination</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>−</mo><mi>c</mi><mo>⋅</mo><mi mathvariant="normal">id</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>⟶</mo><mi>V</mi></mrow><annotation encoding="application/x-tex"> (f - c \cdot \mathrm{id}) \;\colon\; V \longrightarrow V </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</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>-<a class="existingWikiWord" href="/nlab/show/representations">representations</a>. But then, by the first part, this must be an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> or <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero</a>. But it is not an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, by construction, since now the <a class="existingWikiWord" href="/nlab/show/eigenvectors">eigenvectors</a> with <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a> <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> are in the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>. Therefore, all of the linear combination must be <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>−</mo><mi>c</mi><mo>⋅</mo><mi>id</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> f - c \cdot id = 0 </annotation></semantics></math></div> <p>and hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>c</mi><mo>⋅</mo><mi>id</mi></mrow><annotation encoding="application/x-tex"> f = c \cdot id </annotation></semantics></math></div> <p>is a multiple of the identity.</p> </div> </p> <h2 id="InterpretationInCategoricalAlgebra">Interpretation in categorical algebra</h2> <p>The statement of <em>Schur’s lemma</em> is particularly suggestive in the language of <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a>.</p> <p>Here it says that <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a> form a <a class="existingWikiWord" href="/nlab/show/categorification">categorified</a> <em><a class="existingWikiWord" href="/nlab/show/orthogonal+basis">orthogonal basis</a></em> for the <a class="existingWikiWord" href="/nlab/show/2-Hilbert+space">2-Hilbert space</a> of <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional</a> <a class="existingWikiWord" href="/nlab/show/representations">representations</a>, and even an <em><a class="existingWikiWord" href="/nlab/show/orthonormal+basis">orthonormal basis</a></em> if the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> is <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed</a>.</p> <p>After <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> this becomes equivalently the statement that the <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a> form an <em><a class="existingWikiWord" href="/nlab/show/orthogonal+basis">orthogonal basis</a></em> for the <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a>, and even an <em><a class="existingWikiWord" href="/nlab/show/orthonormal+basis">orthonormal basis</a></em> if the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> is <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed</a>.</p> <p>For more on this perspective see also at <em><a class="existingWikiWord" href="/nlab/show/Gram-Schmidt+process">Gram-Schmidt process</a></em> the section <em><a href="Gram-Schmidt+process#CategorifiedGramSchmidtProcess">Categorified Gram-Schmidt process</a></em>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>We now explain this perspective of in more detail:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>Notice that the <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> in a <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Rep</mi></mrow><annotation encoding="application/x-tex">G Rep</annotation></semantics></math> are canonically <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>: given any two <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">f,g \;\colon\; V \to W</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>-<a class="existingWikiWord" href="/nlab/show/representations">representations</a>, also the (value-wise) <a class="existingWikiWord" href="/nlab/show/linear+combination">linear combination</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mi>f</mi><mo>+</mo><msub><mi>c</mi> <mn>2</mn></msub><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">c_1 f + c_2 g \;\colon\; V \to W</annotation></semantics></math> is 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>-homomorphism. (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Rep</mi></mrow><annotation encoding="application/x-tex">G Rep</annotation></semantics></math> is canonically <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>.)</p> <p>Now it makes sense to regard this <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>-valued <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Rep</mi></mrow><annotation encoding="application/x-tex">G Rep</annotation></semantics></math> as analogous</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><msup><mi>Rep</mi> <mi>op</mi></msup><mo>×</mo><mi>G</mi><mi>Rep</mi><mo>⟶</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> hom_G(-,-) \;\colon\; G Rep^{op} \times G Rep \longrightarrow Vect </annotation></semantics></math></div> <p>as a <a class="existingWikiWord" href="/nlab/show/categorification">categorified</a> <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> (see at <em><a class="existingWikiWord" href="/nlab/show/2-Hilbert+space">2-Hilbert space</a></em> for more on this).</p> <p>This is a useful perspective even after <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a>:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>,</mo><mi>W</mi><mo>∈</mo><mi>G</mi><msup><mi>Rep</mi> <mi>fin</mi></msup></mrow><annotation encoding="application/x-tex">V, W \in G Rep^{fin}</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional</a> <a class="existingWikiWord" href="/nlab/show/representations">representations</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>dim</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>hom</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> \langle V,W\rangle \;\coloneqq\; dim\big( hom_G(V,W) \big) \;\in\; \mathbb{N} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of the vector space of homomorphism between them (e.g. <a href="#tomDieck09">tom Dieck 09, p. 29</a>).</p> <p>This construction only depends on the <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>, and hence descends to a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><msubsup><mi>Rep</mi> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow> <mi>fin</mi></msubsup><mo>×</mo><mi>G</mi><msubsup><mi>Rep</mi> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow> <mi>fin</mi></msubsup><mo>⟶</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> \langle -,-\rangle \;\colon\; G Rep^{fin}_{/\sim} \times G Rep^{fin}_{/\sim} \longrightarrow \mathbb{N} </annotation></semantics></math></div> <p>on <a class="existingWikiWord" href="/nlab/show/sets">sets</a> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a>. In fact, under <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> and <a class="existingWikiWord" href="/nlab/show/tensor+product+of+representations">tensor product of representations</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><msubsup><mi>Rep</mi> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow> <mi>fin</mi></msubsup></mrow><annotation encoding="application/x-tex">G Rep^{fin}_{/\sim}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/rig">rig</a>, and this pairing is <a class="existingWikiWord" href="/nlab/show/linear+map">linear</a> with respect to the underlying <a class="existingWikiWord" href="/nlab/show/commutative+monoid">additive monoid</a> structure:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>⟨</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>⊕</mo><msub><mi>V</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>W</mi><mo>⟩</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>⟨</mo><msub><mi>V</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>W</mi><mo>⟩</mo></mrow><mo>+</mo><mrow><mo>⟨</mo><msub><mi>V</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>W</mi><mo>⟩</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\langle V_1 \oplus V_2 \,,\, W \right\rangle \;=\; \left\langle V_1 \,,\, W \right\rangle + \left\langle V_2 \,,\, W \right\rangle </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>⟨</mo><mi>V</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>W</mi> <mn>1</mn></msub><mo>⊕</mo><msub><mi>W</mi> <mi>w</mi></msub><mo>⟩</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>⟨</mo><mi>V</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>W</mi> <mn>1</mn></msub><mo>⟩</mo></mrow><mo>+</mo><mrow><mo>⟨</mo><mi>V</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>W</mi> <mn>2</mn></msub><mo>⟩</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\langle V \,,\, W_1 \oplus W_w \right\rangle \;=\; \left\langle V \,,\, W_1 \right\rangle + \left\langle V \,,\, W_2 \right\rangle </annotation></semantics></math></div> <p>(This is, ultimately, due to the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> as a <a class="existingWikiWord" href="/nlab/show/biproduct">biproduct</a>.)</p> <p>To further strengthen the emerging picture, we may consider the <a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a> of the <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><msubsup><mi>Rep</mi> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow> <mi>fin</mi></msubsup></mrow><annotation encoding="application/x-tex">G Rep^{fin}_{/\sim}</annotation></semantics></math> by passing to its <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, in fact its <a class="existingWikiWord" href="/nlab/show/Grothendieck+ring">Grothendieck ring</a> if we remember also the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+representations">tensor product of representations</a>. This <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> is called the <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mrow><mo>(</mo><mi>G</mi><msubsup><mi>Rep</mi> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow> <mi>fin</mi></msubsup><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> R(G) \;\coloneqq\; K\left( G Rep^{fin}_{/\sim} \right) </annotation></semantics></math></div> <p>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>. By the evident <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-linear extension, the above pairing gives an actual symmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear</a> <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(G)</annotation></semantics></math>:</p> <div class="maruku-equation" id="eq:DecategorifiedInnterProduct"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \langle -,- \rangle \;\colon\; R(G) \times R(G) \longrightarrow \mathbb{Z} </annotation></semantics></math></div> <p>Now the underlying <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(G)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> whose canonical <a class="existingWikiWord" href="/nlab/show/generators+and+relations">generators</a> are nothing but the <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional</a> <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><msub><mo>≃</mo> <mi>ℤ</mi></msub><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">{</mo><msub><mi>V</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R(G) \;\simeq_{\mathbb{Z}}\; \mathbb{Z}\big[ \{V_i\}_i \big] \,. </annotation></semantics></math></div> <p>In summary, in the language of <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, the <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>V</mi> <mi>i</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[V_i]</annotation></semantics></math> constitute a canonical <em><a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a></em> of the <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a>.</p> <p>In terms of this language, <strong>Schur’s lemma</strong> becomes the following statement:</p> <p>The canonical <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> of the <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> given by the <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a> is</p> <ol> <li> <p>generally: an <em><a class="existingWikiWord" href="/nlab/show/orthogonal+basis">ortho-gonal basis</a></em>;</p> </li> <li> <p>even an <em><a class="existingWikiWord" href="/nlab/show/orthonormal+basis">ortho-normal basis</a></em> if the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> is <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a></p> </li> </ol> <p>with respect to the canonical <a class="existingWikiWord" href="/nlab/show/decategorification">decategorified</a> <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> from <a class="maruku-eqref" href="#eq:DecategorifiedInnterProduct">(1)</a>.</p> <h2 id="GeneralizationsAndVariants">Generalizations and variants</h2> <h3 id="for_simple_modules">For simple modules</h3> <p>Part <a href="#FirstItem">(1)</a> of Schur’s lemma is essentially <a class="existingWikiWord" href="/nlab/show/category+theory">category-theoretic</a> and can be generalized in many ways, for example, by replacing the <a class="existingWikiWord" href="/nlab/show/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> by some <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/associative+algebra">algebra</a> and taking the representations compatible with the action of <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>.</p> <h3 id="for_simple_objects_in_an_abelian_category">For simple objects in an abelian category</h3> <p>More generally, part <a href="#FirstItem">(1)</a> of Schur’s lemma applies to <a class="existingWikiWord" href="/nlab/show/simple+objects">simple objects</a> in any <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, and the <a class="existingWikiWord" href="/nlab/show/endomorphism+ring">endomorphism ring</a> of such a <a class="existingWikiWord" href="/nlab/show/simple+object">simple object</a> is a <a class="existingWikiWord" href="/nlab/show/division+ring">division ring</a>, as proved <a href="simple+object#in_an_abelian_category">here</a>.</p> <p>For <a href="#SecondItem">(2)</a>, if the <a class="existingWikiWord" href="/nlab/show/endomorphism+rings">endomorphism rings</a> of all objects in an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> are <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional</a> over an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed 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>, then the endomorphism ring of a simple object is <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> itself, as proved <a href="simple+object#in_an_abelian_category">here</a>. This is the case for the category of finite-dimensional complex representations of a group.</p> <h3 id="for_bridgeland_stable_objects">For Bridgeland stable objects</h3> <p>The statement of Schur’s lemma applies also to objects which are stable with respect to a <a class="existingWikiWord" href="/nlab/show/Bridgeland+stability+condition">Bridgeland stability condition</a>, see <a href="Bridgeland+stability+condition#SchurLemma">there</a>.</p> <h2 id="references">References</h2> <p>Named after <em><a class="existingWikiWord" href="/nlab/show/Issai+Schur">Issai Schur</a></em>.</p> <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: Prop. 1.16 & Cor. 1.17 in: <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>, §1.1.2 in: <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> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Schur%27s_lemma">Schur’s lemma</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 10, 2024 at 20:41:03. 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