The Character of a Group Representation

# The Character of a Group Representation

 Definition: Let $G$ be a group and let $(V, \rho)$ be a group representation of $G$. The Character of $(V, \rho)$ is the function $\chi_V : G \to \mathbb{C}$ defined for all $g \in G$ by $\chi_V(g) = \mathrm{trace} (\rho(g))$.

Recall that if $A$ is a square matrix then $\mathrm{trace} (A)$ is the sum of the main diagonal entries of $A$.

## Example 1

Consider the symmetric group $S_3 = \langle g, r : g^3 = e, r^2 = e, rgr = g^2 \rangle$ and let consider the $2$-dimensional representation $\rho$ of $S_3$ specified by the generators of $G$ by:

(1)
\begin{align} \quad \rho(g) := \begin{bmatrix} -1 & -1 \\ 1 & 0 \end{bmatrix} \quad \begin{bmatrix} -1 & -1 \\ 0 & 1 \end{bmatrix} \end{align}

Then we have that:

(2)
\begin{align} \quad \rho(e) &= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \\ \rho(g^2) &= \begin{bmatrix} 0 & 1 \\ -1 & -1 \end{bmatrix} \\ \rho(gr) &= \begin{bmatrix} 1 & 0 \\ -1 & -1 \end{bmatrix} \\ \rho(rg) &= \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \end{align}

So we see that:

(3)
 Proposition 1: Let $G$ be a finite group and let $(V, \rho_V)$ and $(W, \rho_W)$ be group representations of $G$. Then: a) $\chi_V$ is conjugation invariant, that is, $\chi_V(g) = \chi_V(hgh^{-1})$ for all $g, h \in G$. b) $\chi_V(e) = \mathrm{dim}(V)$ where $e \in G$ denotes the identity element of $G$. c) $\chi_V(g^{-1}) = \overline{\chi_V(g)}$ for all $g \in G$.
The above example is a $2$-dimensional representation of $S_3$ and we calculated that $\chi(e) = 2$, which was to be expected by Proposition 1.b above.
 Proposition 2: Let $G$ be a finite group and let $(V, \rho_V)$ and $(W, \rho_W)$ be group representations of $G$. Then: a) $\chi_{V^*}(g) = \chi_{V}(g^{-1})$ for all $g \in G$. b) $\chi_{V \oplus W} = \chi_V + \chi_W$. c) $\chi_{V \otimes W} = \chi_V \cdot \chi_W$.
Recall that if $(V, \rho_V)$ is a group representation, then the corresponding dual group representation is $(V^*, \rho_V^*)$ where $V^* = \mathrm{How}(V, \mathbb{C})$ is the space of all linear functionals on $V$, and $\rho_V^*$ is defined for all $f \in V^*$ by $\rho_V^*(g)(f)(v) = [f \circ \rho(g)](v)$.
Also recall that if $(V, \rho_V)$ and $(W, \rho_W)$ are group representations of $G$ then $(V \otimes W, \rho_V \oplus \rho_W)$ is another group representation of $G$ where $V \oplus W$ is the direct sum space of $V$ and $W$, and $\rho_V \oplus \rho_W$ is defined via the block matrix $(\rho_V \oplus \rho_W)(g) = \begin{bmatrix} \rho_V(g) & \mathbf{0} \\ \mathbf{0} & \rho_W(g) \end{bmatrix}$.