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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{representation ring} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory} [[!include representation theory - contents]] \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak \noindent\hyperlink{AsEquivariantKTheoryOfThePoint}{As equivariant K-theory of the point}\dotfill \pageref*{AsEquivariantKTheoryOfThePoint} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_the_character_ring}{Relation to the character ring}\dotfill \pageref*{relation_to_the_character_ring} \linebreak \noindent\hyperlink{relation_to_schurs_lemma}{Relation to Schur's lemma}\dotfill \pageref*{relation_to_schurs_lemma} \linebreak \noindent\hyperlink{relation_to_equivariant_ktheory}{Relation to equivariant K-theory}\dotfill \pageref*{relation_to_equivariant_ktheory} \linebreak \noindent\hyperlink{LambdaRingStructure}{Lambda-ring structure}\dotfill \pageref*{LambdaRingStructure} \linebreak \noindent\hyperlink{relation_to_the_burnside_ring}{Relation to the Burnside ring}\dotfill \pageref*{relation_to_the_burnside_ring} \linebreak \noindent\hyperlink{splitting_principle_and_brauer_induction_theorem}{Splitting principle and Brauer induction theorem}\dotfill \pageref*{splitting_principle_and_brauer_induction_theorem} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{spin_group}{Spin group}\dotfill \pageref*{spin_group} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{adams_operations}{Adams operations}\dotfill \pageref*{adams_operations} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Any [[group]] $G$ has a [[category]] of finite-dimensional [[complex number|complex]]-linear [[representations]], often denoted $Rep(G)$. This is a [[symmetric monoidal category|symmetric monoidal]] [[abelian category]] (a ``[[tensor category]]'') and thus has a [[Grothendieck ring]], which is called the \textbf{representation ring} of $G$ and denoted $R(G)$. Elements of the representation ring are hence formal differences (with respect to [[direct sum]]) of ordinary representations: [[virtual representations]]. \hypertarget{in_components}{}\subsubsection*{{In components}}\label{in_components} More concretely, we get $R(G)$ as follows. It has a [[basis]] $(e_i)_i$ given by the [[isomorphism classes]] of \textbf{[[irreducible representations]]} of $G$: that is, $i$ is an index for an irreducible finite-dimensional complex representation of $G$. It has a product given by \begin{displaymath} e_i e_j = \sum_k m_{i j}^k e_k , \end{displaymath} where $m_{i j}^k$ is the multiplicity of the $k$th irrep in the [[tensor product of representations]] of the $i$th and $j$th irreps. In [[physics]] these coefficients are also known as \emph{[[Clebsch-Gordan coefficients]]} (specifically for $G$ the [[special orthogonal group]] $SO(3)$), and the relations they satisfy are also known as \emph{[[Fierz identities]]} (specifically for $G$ a [[spin group]]). Notice that $R(G)$ is commutative thanks to the [[symmetric monoidal category|symmetry]] of the tensor product. \hypertarget{AsEquivariantKTheoryOfThePoint}{}\subsubsection*{{As equivariant K-theory of the point}}\label{AsEquivariantKTheoryOfThePoint} Equivalently the representation ring of $G$ over the [[complex numbers]] is the $G$-[[equivariant K-theory]] of the point, or equivalently by the [[Green-Julg theorem]], if $G$ is a [[compact Lie group]], the [[operator K-theory]] of the [[group algebra]] (the [[groupoid convolution algebra]] of the [[delooping]] groupoid of $G$): \begin{displaymath} R_{\mathbb{C}}(G) \simeq KU^0_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,. \end{displaymath} The first [[isomorphism]] here follows immediately from the elementary definition of equivariant [[topological K-theory]], since a $G$-[[equivariant vector bundle]] over the point is manifestly just a [[linear representation]] of $G$ on a [[complex vector space]]. (e.g. \hyperlink{Greenlees05}{Greenlees 05, section 3}, \hyperlink{Wilson16}{Wilson 16, example 1.6 p. 3}) Therefore a similar isomorphism identifies the $G$-representation ring over the [[real numbers]] with the equivariant orthogonal $K$-theory of the point in degree 0: \begin{displaymath} R_{\mathbb{R}}(G) \;\simeq\; KO_G^0(\ast) \,. \end{displaymath} But beware that equivariant [[KO]], even of the point, is much richer in higher degree (\hyperlink{Wilson16}{Wilson 16, remark 3.34}) In fact, [[equivariant KO-theory]] of the point subsumes the [[representation rings]] over the [[real numbers]], the [[complex numbers]] and the [[quaternions]]: \begin{displaymath} KO_G^n(\ast) \;\simeq\; \left\{ \itexarray{ 0 &\vert& n = 7 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5 \\ R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) } &\vert& n = 4 \\ 0 &\vert& n = 3 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2 \\ R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1 \\ R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0 } \right. \end{displaymath} (\hyperlink{Greenlees05}{Greenlees 05, p. 3}) [[!include Segal completion -- table]] \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{relation_to_the_character_ring}{}\subsubsection*{{Relation to the character ring}}\label{relation_to_the_character_ring} If $G$ is a [[finite group]] and we [[tensor product of modules|tensor]] $R(G)$ with the [[complex numbers]], it becomes [[isomorphism|isomorphic]] to the \textbf{[[character ring]]} of $G$: that is, the ring of complex-valued functions on $G$ that are constant on each [[conjugacy class]]. Such functions are called \textbf{[[class functions]]}. Similarly for $G$ a [[compact Lie group]], its complex linear [[representations]] $\rho \colon G \to U(n) \to Aut(\mathbb{C}^n)$ (for all $n \in \mathbb{N}$) are uniquely specified by their [[characters]] $\chi_\rho \coloneqq tr(\rho(-)) \colon G \to \mathbb{C}$. Therefore also here the representation ring is often called the \emph{character ring} of the group. \hypertarget{relation_to_schurs_lemma}{}\subsubsection*{{Relation to Schur's lemma}}\label{relation_to_schurs_lemma} The ([[isomorphism classes]]) of [[finite-dimensional vector space|finite-dimensional]] [[irreducible representations]] $V_i$ form the canonical $\mathbb{Z}$-[[linear basis]] of the representation ring: \begin{displaymath} R(G) \;\simeq\; \mathbb{Z}\big[ \{V_i\}_i \big] \,. \end{displaymath} Moreover, the [[function]] assigning [[dimensions]] of [[hom-object|hom]]-[[vector spaces]] constitutes a canonical bilinear symmetric [[inner product]]: \begin{displaymath} \langle -,-\rangle \;\colon\; R(G) \times R(G) \longrightarrow \mathbb{Z} \,. \end{displaymath} In terms of this, \emph{[[Schur's lemma]]} states that the [[irreducible representations]] generally constitute an \emph{[[orthogonal basis]]}, and even an \emph{[[orthonormal basis]]} when the [[ground field]] is [[algebraically closed field|algebraically closed]]. For more discussion of this perspective, see \begin{itemize}% \item at \emph{[[Schur's lemma]]} the section \emph{\href{Schur's+lemma#InterpretationInCategoricalAlgebra}{In terms of categorical algebra}}; \item at \emph{[[Gram-Schmidt process]]} the section \emph{\href{Gram-Schmidt+process#CategorifiedGramSchmidtProcess}{Categorified Gram-Schmidt process}}. \end{itemize} \hypertarget{relation_to_equivariant_ktheory}{}\subsubsection*{{Relation to equivariant K-theory}}\label{relation_to_equivariant_ktheory} The representation ring of a [[compact Lie group]] is equivalent to the $G$-[[equivariant K-theory]] of the point. \begin{displaymath} Rep(G) \simeq K_G(\ast) \,. \end{displaymath} The construction of representations by [[index]]-constructions of $G$-equivariant [[Dirac operators]] ([[push-forward in generalized cohomology|push-forward]] in $G$-[[equivariant K-theory]] to the point) is called \emph{[[Dirac induction]]}. [[!include Segal completion -- table]] \hypertarget{LambdaRingStructure}{}\subsubsection*{{Lambda-ring structure}}\label{LambdaRingStructure} The [[Adams operations]] equip the representation ring with the structure of a [[Lambda ring]],. (e.g. \hyperlink{tomDieck79}{tomDieck 79, section 3.5}, \hyperlink{tomDieck09}{tom Dieck 09, section 6.2}, \hyperlink{Meir17}{Meir 17}). \hypertarget{relation_to_the_burnside_ring}{}\subsubsection*{{Relation to the Burnside ring}}\label{relation_to_the_burnside_ring} Let $G$ be a [[finite group]]. Consider \begin{enumerate}% \item the [[Burnside ring]] $A(G)$, which is the [[Grothendieck group]] of the [[monoidal category]] $G Set$ of [[finite set|finite]] [[G-sets]]; \item the [[representation ring]] $R(G)$, which is the [[Grothendieck group]] of the monoidal category $G Rep$ of [[finite dimensional vector space|finite dimensional]] $G$-[[linear representations]]. \end{enumerate} Then then map that sends a G-set to the corresponding linear [[permutation representation]] is a [[strong monoidal functor]] \begin{displaymath} G Set \overset{\mathbb{C}[-]}{\longrightarrow} G Rep \end{displaymath} and hence induces a [[ring homomorphism]] \begin{displaymath} A(G) \overset{ \mathbb{C}[-] }{\longrightarrow} R(G) \end{displaymath} Under the identitification \begin{enumerate}% \item of the [[Burnside ring]] with the [[equivariant stable cohomotopy]] of the point \begin{displaymath} A(G) \;\simeq\; \mathbb{S}_G(\ast) \end{displaymath} (see \href{Burnside+ring#AsTheEquivariantStableCohomotopyOfThePoint}{there}) \item of the [[representation ring]] with the [[equivariant K-theory]] of the point \begin{displaymath} R(G) \;\simeq\; K_G(\ast) \end{displaymath} (as \hyperlink{AsEquivariantKTheoryOfThePoint}{above}) \end{enumerate} this should be image of the initial morphism of [[E-infinity ring spectra]] \begin{displaymath} \mathbb{S} \longrightarrow KU \end{displaymath} from the [[sphere spectrum]] to [[KU]]. \hypertarget{splitting_principle_and_brauer_induction_theorem}{}\subsubsection*{{Splitting principle and Brauer induction theorem}}\label{splitting_principle_and_brauer_induction_theorem} The [[Brauer induction theorem]] says that, over the [[complex numbers]], the representation ring is generated already from the [[induced representations]] of 1-dimensional representations. This may be regarded as the [[splitting principle]] for linear representations and for [[characteristic classes of linear representations]] (\hyperlink{Symonds91}{Symonds 91}). \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{spin_group}{}\subsubsection*{{Spin group}}\label{spin_group} The completion of $Rep(Spin(2k))$ is $\mathbb{Z}[ [ e^{\pm x_j} ] ]$ for $1 \leq j \leq k$ (e.g \hyperlink{Brylinski90}{Brylinski 90, p. 9}). \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Clebsch-Gordan coefficients]], [[Fierz identities]] \item [[Deligne's theorem on tensor categories]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Lecture notes include \begin{itemize}% \item [[Tammo tom Dieck]], section 4.4. in \emph{Representation theory}, 2009 (\href{http://www.uni-math.gwdg.de/tammo/rep.pdf}{pdf}) \end{itemize} Exposition in relation to [[equivariant K-theory]] includes \begin{itemize}% \item [[Akhil Mathew]], \emph{\href{https://amathew.wordpress.com/2011/12/03/equivariant-k-theory/}{Equivariant K-theory}} \item [[John Greenlees]], \emph{Equivariant version of real and complex connective K-theory}, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. (\href{http://projecteuclid.org/euclid.hha/1139839291}{Euclid:1139839291}) \item [[Dylan Wilson]], \emph{Equivariant K-theory}, 2016 (\href{https://www.math.uchicago.edu/~dwilson/notes/equivariant-k-theory-talk.pdf}{pdf}, [[WilsonKTheory16.pdf:file]]) \end{itemize} Classical results for [[compact Lie groups]]: \begin{itemize}% \item [[Graeme Segal]], \emph{The representation ring of a compact Lie group}, Publications Math\'e{}matiques de l'Institut des Hautes \'E{}tudes Scientifiques, January 1968, Volume 34, Issue 1, pp 113-128 (\href{http://archive.numdam.org/numdam-bin/fitem?id=PMIHES_1968__34__113_0}{NUMDAM}) \item Masaru Tackeuchi, \emph{A remark on the character ring of a compact Lie group}, J. Math. Soc. Japan Volume 23, Number 4 (1971), 555-705 (\href{http://projecteuclid.org/euclid.jmsj/1259849785}{Euclid}) \end{itemize} In the generality of [[super Lie groups]]: \begin{itemize}% \item [[Gregory Landweber]], \emph{Representation rings of Lie superalgebras}, K-Theory 36 (2005), no. 1-2, 115-168 (\href{http://arxiv.org/abs/math/0403203}{arXiv:math/0403203}) \item [[Gregory Landweber]], \emph{Twisted representation rings and Dirac induction}, J. Pure Appl. Algebra 206 (2006), no. 1-2, 21-54 (\href{http://arxiv.org/abs/math/0403524}{arXiv:math/0403524}) \end{itemize} With an eye towards [[loop group representations]]: \begin{itemize}% \item [[Jean-Luc Brylinski]], \emph{Representations of loop groups, Dirac operators on loop space, and modular forms}, Topology, 29(4):461--480, 1990. \end{itemize} Concerning the [[Brauer induction theorem]] and the [[splitting principle]]: \begin{itemize}% \item [[Peter Symonds]], \emph{A splitting principle for group representations}, Comment. Math. Helv. (1991) 66: 169 (\href{https://eudml.org/doc/140229}{dml:140229}, \href{https://doi.org/10.1007/BF02566643}{doi:10.1007/BF02566643}) \end{itemize} \hypertarget{adams_operations}{}\subsubsection*{{Adams operations}}\label{adams_operations} The [[Adams operations]] and [[Lambda-ring]]-structure on representation rings are discussed in \begin{itemize}% \item [[Tammo tom Dieck]], section 3.5 of \emph{[[Transformation Groups and Representation Theory]]}, Lecture Notes in Mathematics 766 Springer 1979 \item Robert Boltje, \emph{A characterization of Adams operations on representation rings}, 2001 (\href{https://boltje.math.ucsc.edu/publications/p01v.pdf}{pdf}) \item [[Tammo tom Dieck]], section 6.2 of \emph{Representation theory}, 2009 (\href{http://www.uni-math.gwdg.de/tammo/rep.pdf}{pdf}) \item [[Michael Boardman]], \emph{Adams operations on Group representations}, 2007 (\href{http://www.math.jhu.edu/~jmb/note/adamrept.pdf}{pdf}) \item Ehud Meir, [[Markus Szymik]], \emph{Adams operations and symmetries of representation categories} (\href{https://arxiv.org/abs/1704.03389}{arXiv:1704.03389}) \end{itemize} [[!redirects representation rings]] \end{document}