Character of a representation

Definition

Let $G$ be a finite group and let $\rho:G\to \mathrm{GL}(V)$ be a finite-dimensional complex representation . The character of $\rho$ is the function

where $\mathrm{tr}$ is the trace of a linear operator on $V$.

Basic properties

1. Class function. For all $g,h\in G$,

   $$\chi_\rho(hgh^{-1})=\chi_\rho(g).$$

   Equivalently, $\chi_\rho$ is constant on conjugacy classes , i.e. it is a class function .

2. Isomorphism invariance. If $\rho\simeq \rho'$ (isomorphic representations), then $\chi_\rho=\chi_{\rho'}$.

3. Additivity and multiplicativity.

   • For a direct sum $\rho\oplus \sigma$, $\chi_{\rho\oplus \sigma}=\chi_\rho+\chi_\sigma.$
   • For a tensor product $\rho\otimes \sigma$, $\chi_{\rho\otimes \sigma}(g)=\chi_\rho(g)\,\chi_\sigma(g).$

4. Dimension. $\chi_\rho(e)=\dim_\mathbb{C}V$.

Characters are central tools because many structural questions about representations reduce to identities among class functions and the orthogonality relations .

Examples

Example 1: Trivial representation

If $\rho$ is the trivial representation on $V$ (every $g$ acts as $\mathrm{id}_V$), then

for all $g\in G$.

Example 2: Regular representation

Let $\mathbb{C}[G]$ be the group algebra and let $G$ act by left multiplication (the regular representation ). Its character satisfies

(For $g\neq e$, left multiplication by $g$ permutes the basis $\{\delta_h\}$ without fixed points, so the corresponding permutation matrix has trace $0$.)

Example 3: The standard $2$-dimensional representation of $S_3$

Let $S_3$ act on $\mathbb{C}^3$ by permuting coordinates, and restrict to the $2$-dimensional subspace

which is $S_3$-stable. The resulting representation $\rho_{\mathrm{std}}$ has character values (constant on conjugacy classes):

• $\chi_{\mathrm{std}}(e)=2$,
• $\chi_{\mathrm{std}}(\text{a transposition})=0$,
• $\chi_{\mathrm{std}}(\text{a 3-cycle})=-1$. This is the character of the unique $2$-dimensional irreducible representation of $S_3$.