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\begin{document}

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\section*{group character}

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

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

[[!include representation theory - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - 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{Weights}{Restriction of group characters to maximal tori -- weights}\dotfill \pageref*{Weights} \linebreak
\noindent\hyperlink{RelationToChernRootsAndSplittingPrinciple}{Relation to Chern roots and the splitting principle}\dotfill \pageref*{RelationToChernRootsAndSplittingPrinciple} \linebreak
\noindent\hyperlink{CharactersAndFundamentalGroupsOfTori}{Characters and fundamental group of tori}\dotfill \pageref*{CharactersAndFundamentalGroupsOfTori} \linebreak
\noindent\hyperlink{inner_product_and_orthogonality}{Inner product and orthogonality}\dotfill \pageref*{inner_product_and_orthogonality} \linebreak
\noindent\hyperlink{in_terms_of_the_classifying_space_of_the_group}{In terms of the classifying space of the group}\dotfill \pageref*{in_terms_of_the_classifying_space_of_the_group} \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}

A \emph{multiplicative [[character]]} of a [[group]] $G$ is a [[group homomorphism]] into the [[circle group]] $U(1)$, or more generally into the [[group of units]] $k^\times$ of a given [[ground field]] (for instance $\mathbb{C}^\times = U(1)$):

\begin{displaymath}
\chi
  \;\colon\;
  G \longrightarrow k^\times  
  \,.
\end{displaymath}
Since $k^\times$ is an [[abelian group]], this means that group characters are in particular [[class functions]].

Dually a co-character is a homomorphism out of $k^\times$ into $G$.

The collection of characters is itself an [[abelian group]] under the pointwise multiplication, this is called the \textbf{character lattice} $Hom(G,k^\times)$ of the group. Similarly the \textbf{cocharacter lattice} is $Hom(k^\times, G)$.

For [[topological groups]] one considers [[continuous map|continuous]] characters. Specifically, for a [[locally compact Hausdorff]] group $G$ (often further assumed to be an [[abelian group]]), a \textbf{character} of $G$ is continuous homomorphism to the [[circle]] group $\mathbb{R}/\mathbb{Z}$. If $G$ is [[profinite group|profinite]], then this is the same as an continuous homomorphism to the [[discrete space|discrete]] group $\mathbb{Q}/\mathbb{Z}$. (See \href{http://mathoverflow.net/questions/86089/two-definitions-of-character-of-topological-groups}{MO}.)

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

\hypertarget{Weights}{}\subsubsection*{{Restriction of group characters to maximal tori -- weights}}\label{Weights}

Let $G$ be a [[connected topological space|connected]] [[compact Lie group]].

By the general \href{maximal+torus#Properties}{properties of maximal tori} in this case, it follows that every group character $G \to U(1)$ is already fixed by its restriction along a [[maximal torus]] inclusion

\begin{displaymath}
T \hookrightarrow G \to U(1)
  \,.
\end{displaymath}
Now the group characters of the [[abelian group|abelian]] maximal torus $T\simeq U(1)^n$ are [[n-tuples]] of group characters of the [[circle group]] $U(1)$, which are [[integers]] -- the \emph{[[weight (in representation theory)|weights]]}.

Explicitly, under a given identification of the [[circle group]] as a [[quotient]] of the additive group of [[real numbers]]

\begin{displaymath}
U(1) \simeq \mathbb{R}/h \mathbb{Z}
\end{displaymath}
for $h \in (0,\infty)$, then the character $\lambda$ on $U(1)^n\simeq T$ labeled by $(x_1, \cdots, x_n) \in \mathbb{Z}^n$ is

\begin{displaymath}
\lambda(t_1,\cdots t_n) 
  = 
  \exp(\tfrac{i}{\hbar} \sum_{j = 1}^n t_j n_j )
\end{displaymath}
(where $\hbar \coloneqq h/2\pi$ is ``[[Planck's constant]]'').

(e.g. \hyperlink{Johansen}{Johansen, section 2.10})

\hypertarget{RelationToChernRootsAndSplittingPrinciple}{}\subsubsection*{{Relation to Chern roots and the splitting principle}}\label{RelationToChernRootsAndSplittingPrinciple}

A group character, hence a group [[homomorphism]] $G \to U(1)$ induces a map of [[classifying spaces]] $B G \to B U(1) \simeq K(\mathbb{Z},2)$. Similarly for the restriction to the [[maximal torus]] \hyperlink{Weights}{above}, which induces

\begin{displaymath}
B U(1)^n \simeq B T \to B G \to B U(1)\simeq K(\mathbb{Z},2)
  \,.
\end{displaymath}
Under this identification the \hyperlink{Weights}{weights} $x_i$ of the group character, as above, are the ``[[Chern roots]]'' as the appear in the [[splitting principle]]. See there for more.

\hypertarget{CharactersAndFundamentalGroupsOfTori}{}\subsubsection*{{Characters and fundamental group of tori}}\label{CharactersAndFundamentalGroupsOfTori}

Write $S^1$ for the [[circle group]].

Let $T$ be a [[torus]], regarded as an [[abelian group]]. Write $[T,S^1]$ for its character group.

There is a [[bilinear form]]

\begin{displaymath}
\pi_1(T)\otimes [T, S^1] \longrightarrow \mathbb{Z}
\end{displaymath}
on the [[fundamental group]] of the [[torus]] and its character group, given by sending a [[homotopy class]] $[\gamma]$ of a [[continuous map]]

\begin{displaymath}
\gamma \colon S^1 \longrightarrow T
\end{displaymath}
to the [[homotopy class]] $c(\gamma)$ of the [[composition]] with a [[character]] $c \colon T \longrightarrow S^1$

\begin{displaymath}
c(\gamma) \;\colon\; S^1 \stackrel{\gamma}{\longrightarrow}
  T \stackrel{c}{\longrightarrow} S^1
\end{displaymath}
regarded as an element $[c(\gamma)] \in \pi_1(S^1) \simeq \mathbb{Z}$.

This [[bilinear form]] is non-degenerate, and hence constitutes an [[isomorphism]]

\begin{displaymath}
\pi_1(T) \simeq [T,S^1]
  \,.
\end{displaymath}
\hypertarget{inner_product_and_orthogonality}{}\subsubsection*{{Inner product and orthogonality}}\label{inner_product_and_orthogonality}

The complex [[class functions]] on a [[finite group]] $G$ have an [[inner product]] given by

\begin{displaymath}
\langle \alpha, \beta\rangle
  \coloneqq
  \frac{1}{{\vert G \vert}}
  \underset{g \in G}{\sum} \alpha(g) \overline{\beta(g)}
  \,.
\end{displaymath}
The \emph{[[Schur orthogonality relation]]} is the statement that the [[irreducible representation|irreducible]] group characters $\{\chi_i\}_i$ form an [[orthonormal basis]] of the space of [[class functions]] under this [[inner product]]:

\begin{displaymath}
\langle \chi_i, \chi_j \rangle 
  = 
  \left\{
    \itexarray{
       1 & if \; i = j
       \\
       0 & otherwise
    }
  \right.
\end{displaymath}
Such properties arise from characters occurring as [[traces]] of \href{representation+ring#relation_to_the_character_ring}{group representations}.

\hypertarget{in_terms_of_the_classifying_space_of_the_group}{}\subsubsection*{{In terms of the classifying space of the group}}\label{in_terms_of_the_classifying_space_of_the_group}

Consider the [[classifying space]], $B G$, of the group. Then its [[free loop space]], $Map (S^1, B G)$, has as components $G$ modulo [[conjugation]]. Then, the characters of $G$ may be expressed as the zeroth cohomology of this loop space, $H^0(\mathcal{L} B G, \mathbb{C})$. This construction is useful in the generalisation to [[transchromatic characters]].

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

\begin{itemize}%
\item [[Pontryagin duality]], [[Pontryagin dual]]


\item [[conjugacy class of subgroups]]



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

Original articles on character rings/[[representation rings]] of [[compact Lie groups]] include

\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}
Lecture notes on characters of finite and compact Lie groups include

\begin{itemize}%
\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}


\item Andrei Yafaev, \emph{Characters of finite groups} (\href{http://www.ucl.ac.uk/~ucahaya/Characters.pdf}{pdf})



\end{itemize}
Discussion for finite groups 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}
[[!redirects group characters]]

[[!redirects character lattice]] [[!redirects character lattices]]

[[!redirects cocharacter lattice]] [[!redirects cocharacter lattices]]

[[!redirects co-character lattice]] [[!redirects co-character lattices]]

[[!redirects character group]] [[!redirects character groups]]



\end{document}