Group representation

A linear action of a group on a vector space, equivalently a homomorphism into a general linear group.

Group representation

Definition

Let $G$ be a group and let $k$ be a field. A (linear) representation of $G$ over $k$ is a pair $(V,\rho)$ where

Equivalently, $G$ acts on $V$ by $k$ -linear automorphisms via

g\cdot v := \rho(g)(v),

so that $e\cdot v = v$ and $(gh)\cdot v = g\cdot(h\cdot v)$ for all $g,h\in G$ , $v\in V$ .

The dimension (or degree) of the representation is $\dim_k(V)$ .

A representation $(V,\rho)$ is equivalently a left module over the group algebra $k[G]$ : extend $\rho$ $k$ -linearly to an algebra homomorphism

k[G]\longrightarrow \mathrm{End}_k(V),

where $\mathrm{End}_k(V)$ consists of linear maps $V\to V$ .

A homomorphism of representations $f:(V,\rho)\to (W,\sigma)$ is a $k$ -linear map $f:V\to W$ such that

f(\rho(g)v)=\sigma(g)f(v)\quad\text{for all } g\in G,\ v\in V.

Equivalently, $f$ is a $k[G]$ -module homomorphism (compare module homomorphism ).

To any representation one associates its character $\chi_\rho(g)=\mathrm{tr}(\rho(g))$ , using the trace .

Trivial representation (any group).
Take $V=k$ and $\rho(g)=\mathrm{id}_k$ for all $g\in G$ . Then $g\cdot v=v$ for all $g$ , and $\chi(g)=1$ .
Sign representation of $S_3$ .
Over any field $k$ with $\mathrm{char}(k)\neq 2$ , define $\rho:S_3\to k^\times\subset \mathrm{GL}_1(k)$ by $\rho(\sigma)=\mathrm{sgn}(\sigma)$ . This is 1-dimensional and nontrivial.
One-dimensional representations of a cyclic group.
Let $C_n=\langle g\mid g^n=1\rangle$ , and take $k=\mathbb{C}$ . For each integer $j$ , define $\rho_j:C_n\to \mathbb{C}^\times$ by $\rho_j(g)=\zeta_n^j$ , where $\zeta_n=e^{2\pi i/n}$ . Then $\rho_j(g^m)=\zeta_n^{jm}$ and $\dim(\rho_j)=1$ .