Character of a linear representation

This term makes sense in the context of a linear representation of a group, viz an action of the group as linear automorphisms of a vector space

This article gives a basic definition in the following area: linear representation theory
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Definition

Definition in terms of linear representation as a homomorphism

Let $G$ be a group and $\rho :G\to GL(V)$ be a finite-dimensional linear representation over a field $k$. Then, the character of $\rho$ is the composite $Tr\circ \rho$ where $Tr$ is the trace map from $GL(V)$ to $k$.

Elementary properties

Characters of representations are class functions, that is, they are constant on each conjugacy class of the group.

A character is called irreducible if its corresponding representation is an irreducible representation.

Character tables

The notion of characters leads to that of a character table of a group; given a group $G$, the character table lists the value of each of the irreducible characters of the group on each conjugacy class (and thus each element, since the character is a class function)

Examples

• The trivial representation of a group admits the trivial character - equal to $1$ on each element.

• The character of the regular representation of a finite group $G$ takes the value $|G|$ at the identity element and zero elsewhere. This is because the character of the corresponding permutation matrix is the number of fixed points, and multiplication by a non-identity element has no fixed points.

• The sign representation on the symmetric group $S_{n}$ has character $1$ on even elements, $-1$ on odd elements.

• The standard representation of the symmetric group $S_{n}$ has character $\chi (g)$ equal to the number of fixed points of the permutation $g\in S_{n}$ minus one.