Character of a Direct Sum

Let $G$ be a finite group and let $V$ be a finite-dimensional complex representation with action $\rho_V:G\to \mathrm{GL}(V)$. Its character is the class function

using the trace .

If $V$ and $W$ are representations, their direct sum $V\oplus W$ is a direct sum representation with

Proposition

For all $g\in G$,

More generally, for a finite direct sum $\bigoplus_{i=1}^m V_i$,

This identity is used constantly alongside character orthogonality to compute multiplicities of irreducible representations inside a given representation.

Examples

Example 1: $S_3$ permutation representation on 3 letters

Let $V=\mathbb{C}^3$ with $S_3$ acting by permuting the standard basis. Its character counts fixed points:

• $\chi_V(e)=3$,
• for a transposition, $\chi_V(\text{transposition})=1$,
• for a 3-cycle, $\chi_V(\text{3-cycle})=0$.

This representation decomposes as $V \cong \mathbf{1} \oplus V_{\mathrm{std}}$, where $\mathbf{1}$ is the trivial rep and $V_{\mathrm{std}}$ is the 2D standard irreducible. The characters satisfy

with $\chi_{\mathbf{1}}=(1,1,1)$ and $\chi_{\mathrm{std}}=(2,0,-1)$ on the three conjugacy classes, giving $(3,1,0)$ as required.

Example 2: Cyclic group $C_n$

Let $G=C_n=\langle g\rangle$. Fix $\zeta=e^{2\pi i/n}$. For integers $a,b$, let $V_a$ and $V_b$ be 1D representations with $g$ acting by $\zeta^a$ and $\zeta^b$. Then

and the direct sum satisfies

Example 3: Adding a trivial summand

For any $V$, the character of $V\oplus \mathbf{1}$ is $\chi_V + 1$, i.e. $\chi_{V\oplus \mathbf{1}}(g)=\chi_V(g)+1$ for every $g\in G$.