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\begin{document}

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\section*{character}

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

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

[[!include representation theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{character_on_a_group}{Character on a group}\dotfill \pageref*{character_on_a_group} \linebreak
\noindent\hyperlink{CharacterOfALinearRepresentation}{Character of a linear representation}\dotfill \pageref*{CharacterOfALinearRepresentation} \linebreak
\noindent\hyperlink{left_and_right_characters_on_a_ring_over_a_ring}{Left and right characters on a ring over a ring}\dotfill \pageref*{left_and_right_characters_on_a_ring_over_a_ring} \linebreak
\noindent\hyperlink{character_of_a_topological_space}{Character of a topological space}\dotfill \pageref*{character_of_a_topological_space} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

There are many notions of a character for an algebraic structure, often topologized.

\hypertarget{character_on_a_group}{}\subsubsection*{{Character on a group}}\label{character_on_a_group}

A \emph{multiplicative character} on a ([[discrete group|discrete]]) [[group]] $G$ (a \emph{[[group character]]}) is a [[group homomorphism]] from $G$ to the [[group of units]] $k^\times$ of the [[ground field]] $k$:

\begin{displaymath}
\chi
  \;\colon\;
  G 
    \longrightarrow 
  k^\times
\end{displaymath}
Since the [[codomain]] $k^\times$ is an [[abelian group]] we have that

\begin{enumerate}%
\item such a [[group homomorphism]] is [[invariant]] under [[conjugation]], so that a group character descends to a [[function]] on the [[set]] of [[conjugacy classes]] of elements in $G$.


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



\end{enumerate}
For more see at \emph{[[group character]]}.

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 a 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{CharacterOfALinearRepresentation}{}\subsubsection*{{Character of a linear representation}}\label{CharacterOfALinearRepresentation}

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{left_and_right_characters_on_a_ring_over_a_ring}{}\subsubsection*{{Left and right characters on a ring over a ring}}\label{left_and_right_characters_on_a_ring_over_a_ring}

Let $A$ be a unital, not necessarily commutative, [[ring]] (or, more generally, $k$-algebra for $k$-commutative); then a monoid in a category of $A$-bimodules (which are respectively also compatibly $k$-modules), is called an $A$-ring. In other words an $A$-ring $B$ is an object in the [[coslice category]] $A\Ring$; it is thus a ring $B$ equipped with multiplication $\mu_B$ and a map $\eta: A\to B$ of rings.

A \textbf{left character} of an $A$-ring $(B,\mu_B,\eta_B)$ is a map $\chi:B\to A$ such that

(i) (left $A$-linearity) $\chi(\eta(a)b) = a\chi(b)$ for all $a\in A$, $b\in B$

(ii) (associativity) $\chi(b b') = \chi(b (\eta\circ\chi)(b'))$ for all $b,b'\in B$

(iii) (unitality) $\chi(1_B) = \chi(1_A)$

where we denoted multiplication in $A$ and in $B$ by concatenation. The conditions on $\chi$ can be restated as the requirement that the map $B\otimes A\to A$ given by $b\otimes a\mapsto \chi(b\eta(a))$ is a $B$-action extending the left regular $A$-action (i.e. the multiplication on $A$ considered as a left action).

Dually, a \textbf{right character} of an $A$-ring $(B,\mu_B,\eta_B)$ is a map $\chi:B\to A$ such that

(i) (right $A$-linearity) $\chi(b\eta(a)) = \chi(b)a$ for all $a\in A$, $b\in B$

(ii) (associativity) $\chi(b b') = \chi((\eta\circ\chi)(b) b')$ for all $b,b'\in B$

(iii) (unitality) $\chi(1_B) = \chi(1_A)$

This is in turn equivalent to extending the right regular action of $A$ to the action of $B$ on $A$.

\hypertarget{character_of_a_topological_space}{}\subsubsection*{{Character of a topological space}}\label{character_of_a_topological_space}

The character $\chi(X,x)$ of a [[topological space]] $X$ at a point $x$ is the minimal cardinality of a local basis of neighborhoods of point $x$ (local [[topological basis|basis of topology]] on $X$) if it is infinite and aleph zero otherwise. The character of a topological space is the supremum of $\chi(X,x)$ when $x$ runs through $X$.

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

\begin{itemize}%
\item [[characters are cyclotomic integers]]

\end{itemize}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{itemize}%
\item [[character table of 2D4=Dic2=Q8]]


\item [[character table of 2T]]


\item [[character table of 2O]]


\item [[character table of 2I]]


\item [[character table of GL(2,3)]]



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

\begin{itemize}%
\item [[Weyl character formula]]


\item [[Kac character formula]]


\item [[Kac-Weyl character]]


\item [[Cheeger-Simons differential character]]


\item [[Chern root]], [[splitting principle]]


\item [[character ring]]


\item [[table of marks]]



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

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 character]] [[!redirects characters]]

[[!redirects character ring]] [[!redirects character rings]]



\end{document}