Character of a Tensor Product

Let $G$ be a finite group and let $V,W$ be finite-dimensional complex representations with actions $\rho_V,\rho_W$. Their tensor product $V\otimes W$ is a representation via

The character is $\chi_V(g)=\mathrm{tr}(\rho_V(g))$.

Proposition

For all $g\in G$,

Equivalently, $\chi_{V\otimes W} = \chi_V\cdot \chi_W$ as functions $G\to \mathbb{C}$.

This follows from the linear algebra identity

using the trace .

Examples

Example 1: Cyclic group $C_n$

For $G=C_n=\langle g\rangle$, 1D characters satisfy multiplication under tensor product. Let $V_a,V_b$ be 1D reps with $g\mapsto \zeta^a$ and $g\mapsto \zeta^b$ where $\zeta=e^{2\pi i/n}$. Then

so $V_a\otimes V_b \cong V_{a+b}$.

Example 2: $S_3$: tensoring by the sign representation

Let $\varepsilon$ be the 1D sign representation of $S_3$, and let $\sigma$ be the 2D standard irreducible. On the three conjugacy classes $(e),(\text{transposition}),(\text{3-cycle})$,

Then

so $\sigma\otimes \varepsilon \cong \sigma$.

Example 3: $S_3$: $\sigma\otimes\sigma$

Using $\chi_{\sigma\otimes\sigma}=\chi_\sigma^2$ pointwise gives

Decomposing into irreducibles using the known irreducible characters $\mathbf{1}=(1,1,1)$, $\varepsilon=(1,-1,1)$, $\sigma=(2,0,-1)$, we check:

so