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\begin{document}

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\section*{character of a linear representation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_character_homomorphism}{The character homomorphism}\dotfill \pageref*{the_character_homomorphism} \linebreak
\noindent\hyperlink{SchurInnerProduct}{Schur inner product}\dotfill \pageref*{SchurInnerProduct} \linebreak
\noindent\hyperlink{SpecialCharacterValues}{Special character values}\dotfill \pageref*{SpecialCharacterValues} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[representation theory]], one defines the \emph{character of a [[linear representation]] $\rho\colon G\to End(V)$ to be the [[group character]] on $G$ given by $g \mapsto Tr \rho(g)$, whenever the [[trace]] in $V$ makes sense (e.g. when $V$ is [[finite-dimensional vector space|finite-dimensional]]). Since such a function is invariant under conjugation, we may equivalently consider it a function on the set of [[conjugacy classes]] of elements in $G$.}

Sometimes we also extend a character linearly to the free vector space on the set of conjugacy classes. This version of the character can be identified with the [[bicategorical trace]] of the identity map of the representation, considered as a $k[G]$-$k$-module.

There is a different notion of an \emph{infinitesimal character} in [[Harish–Chandra theory]] and also a notion of the \emph{formal character}.

There are important formulas concerning characters in representation theory, like [[Weyl character formula]], Kirillov character formula, Demazure character formula and so on. There is also a formula for the [[induced character]] of an [[induced representation]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{the_character_homomorphism}{}\subsubsection*{{The character homomorphism}}\label{the_character_homomorphism}

\begin{prop}
\label{InCharZeroCharacterMorphismIsInjective}\hypertarget{InCharZeroCharacterMorphismIsInjective}{}
\textbf{(in [[characteristic zero]] the [[character homomorphism]] is [[injective function|injective]])}

For $G$ a [[finite group]] and $k$ a [[field]] of [[characteristic zero]], the [[character homomorphism]]

\begin{displaymath}
\itexarray{
    Rep_k(G)_{/\sim}
    &\overset{\chi}{\longrightarrow}&
    k^{ConjCl(G)}
    \\
    V 
      &\mapsto&
    \big(
      [g] \mapsto \chi_V(g) 
    \mathrlap{
      \coloneqq \mathrm{tr}_V(g)
    \big)
    }
  }
\end{displaymath}
is [[injective function|injective]].

\end{prop}
(e.g. \hyperlink{tomDieck09}{tom Dieck 09, Theorem (2.1.3)})

\hypertarget{SchurInnerProduct}{}\subsubsection*{{Schur inner product}}\label{SchurInnerProduct}

\begin{prop}
\label{SchurInnerProductInTermsOfCharacters}\hypertarget{SchurInnerProductInTermsOfCharacters}{}
\textbf{(\href{Schur's+lemma#InterpretationInCategoricalAlgebra}{Schur inner product} in terms of [[character of a linear representation|characters]])}

Let $G$ be a [[finite group]] of [[order of a group|order]] ${\vert G \vert} \in \mathbb{N}$, and let $V_1, V_2 \in Rep_k(G)$ be two [[linear representation]] over a [[ground field]] $k$ of [[characteristic zero]].

Then the \href{Schur's+lemma#InterpretationInCategoricalAlgebra}{Schur inner product} of the two representation equals the [[convolution product]] of their [[character of a linear representation|characters]], normalized by the [[order of a group|order]] of $G$:

\begin{equation}
\left\langle
    V_1, V_2
  \right\rangle
  \;\coloneqq\;
  dim_k\Big(
    Hom\big(
      V_1, V_2
    \big)
  \Big)
  \;=\;
  \frac{1}{\left\vert G\right\vert}
  \underset{g \in G}{\sum}
  \chi_{V_1}(g^{-1})
    \cdot
  \chi_{V_2}(g)
  \,.
\label{OrthogonalityRelation}\end{equation}
\end{prop}
(e.g. \hyperlink{tomDieck09}{tom Dieck 09 (2.5)})

\begin{example}
\label{NormalizedSumOfCharacters}\hypertarget{NormalizedSumOfCharacters}{}
\textbf{(normalized [[sum]] of [[character of a linear representation|characters]] is [[fixed point space]]-[[dimension]])}

Let $G$ be a [[finite group]] of [[order of a group|order]] ${\vert G \vert} \in \mathbb{N}$, and let $V \in Rep_k(G)$ be a [[linear representation]] over a [[ground field]] $k$ of [[characteristic zero]].

Then the [[dimension]] of the [[fixed point space]] $V^G$ of $V$ under $G$ equals the normalized [[sum]] of the character values over all group elements:

\begin{equation}
dim\left( V^G \right)
  \;=\;
  \frac{1}{{\vert G\vert}}
  \underset{g \in G}{\sum}
  \chi_V(g)
  \,.
\label{SumOverAllCharacterValues}\end{equation}
In particular

\begin{enumerate}%
\item if $V = \rho \neq \mathbf{1}$ is a non-[[trivial representation|trivial]] [[irreducible representation]] then

\end{enumerate}
\begin{displaymath}
\underset{g \in G}{\sum}
    \chi_{\rho}(g)
    \;=\;
    0
\end{displaymath}
\begin{enumerate}%
\item if $V = \mathbf{1}$ is the the [[trivial representation|trivial]] [[irreducible representation]] then

\end{enumerate}
\begin{displaymath}
\underset{g \in G}{\sum}
    \chi_{\mathbf{1}}(g)
    \;=\;
    \left\vert G\right\vert
\end{displaymath}
\end{example}
(e.g. \hyperlink{tomDieck09}{tom Dieck 09 (2.1)})

\begin{proof}
The first statement is a special case of Prop. \ref{SchurInnerProductInTermsOfCharacters} by observing that the [[fixed point space]] of a [[linear representation]] $V$ is the space of [[homomorphisms]] from the [[trivial representation|trivial]] [[irrep]]

\begin{displaymath}
dim\big( V^G\big)
  \;=\;
  dim\Big( 
    \mathrm{Hom}\big( \mathbf{1}, V \big)
  \Big)
\end{displaymath}
and that the [[character of a linear representation|character]] of $\mathbf{1}$ is [[constant function|constant]] on $1 \in k$.

The second statement follows from this by [[Schur's lemma]], which says that $\langle \mathbf{1}, \rho\rangle = 0$ if the [[irrep]] $\rho \neq \mathbf{1}$.

The last statement may similarly be seen as the complementary special case of \eqref{SumOverAllCharacterValues} obtained from $\langle \mathbf{1},\mathbf{1}\rangle =1$, but of course it also follows directly from the fact that the [[character of a linear representation|character]] of the [[trivial representation|trivial]] [[irrep]] is [[constant function|constant]] on 1.

\end{proof}
\hypertarget{SpecialCharacterValues}{}\subsubsection*{{Special character values}}\label{SpecialCharacterValues}

\begin{prop}
\label{CharactersAreCyclotomicIntegers}\hypertarget{CharactersAreCyclotomicIntegers}{}
\textbf{([[characters are cyclotomic integers]])}

Let $G$ be a [[finite group]], and let $V$ be a [[finite-dimensional vector space|finite-dimensional]] [[linear representation]] over a [[ground field]] $k$

Then the values of the [[character of a representation|character]] $\chi_V \colon G \to k$ of $V$ are [[cyclotomic integers]] over $k$, for some [[root of unity]].

\end{prop}
(see e.g. \hyperlink{characters+are+cyclotomic+integers#Naik}{Naik})

In particular it follows that:

\begin{prop}
\label{}\hypertarget{}{}
If the [[ground field]] $k$ in Prop. \ref{CharactersAreCyclotomicIntegers} has [[characteristic zero]], then a [[character of a linear representation]] which takes values in the [[rational numbers]] $\mathbb{Q} \hookrightarrow k$ in fact already takes values in the [[integers]] $\mathbb{Z} \hookrightarrow\mathbb{Q} \hookrightarrow k$.

\end{prop}
(see e.g. \hyperlink{characters+are+cyclotomic+integers#Yang}{Yang})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[equivariant K-theory]]


\item [[fractional D-brane]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Tammo tom Dieck]], chapter 2 of \emph{Representation theory}, 2009 (\href{http://www.uni-math.gwdg.de/tammo/rep.pdf}{pdf})

\end{itemize}
For more see the references at \emph{[[representation theory]]}.

Character rings of [[compact Lie groups]] are discussed in

\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})


\item Troels Roussauc Johansen, \emph{Character Theory for Finite Groups and Compact Lie Groups} \href{http://www.math.upb.de/~johansen/character-theory.pdf}{pdf}



\end{itemize}
Discussion in the more general context of [[equivariant cohomology|equivariant]] [[complex oriented cohomology theory]] ([[transchromatic character]]) is in

\begin{itemize}%
\item [[Michael Hopkins]], [[Nicholas Kuhn]], [[Douglas Ravenel]], \emph{Generalized group characters and complex oriented cohomology theories}, J. Amer. Math. Soc. 13 (2000), 553-594 (\href{http://www.ams.org/journals/jams/2000-13-03/S0894-0347-00-00332-5/}{publisher}, \href{http://www.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf}{pdf})

\end{itemize}
for review see

\begin{itemize}%
\item Arpon Raksit, \emph{Characters in global equivariant homotopy theory}, 2015 \href{http://web.stanford.edu/~arpon/math/files/senior-thesis.pdf}{pdf}

\end{itemize}
Examples of [[characters of linear representations]] of [[finite groups]] are discussed and listed at

\begin{itemize}%
\item [[James Montaldi]], \emph{\href{http://www.maths.manchester.ac.uk/~jm/wiki/Representations/Representations}{representations}}, 2008

\end{itemize}
[[!redirects characters of linear representations]]

[[!redirects character of a representation]] [[!redirects characters of representations]]

[[!redirects character homomorphism]] [[!redirects character homomorphisms]]

[[!redirects character morphism]] [[!redirects character morphisms]]

[[!redirects character table]] [[!redirects character tables]]



\end{document}