Prove that $χ_τ (g_1, g_2) = χ_ρ(g_1)χ_ϕ(g_2)$.

Question

Let $G_1$ and $G_2$ be finite groups and let $G = G_1 ×G_2$. Suppose $ρ: G_1 → GL_m(\Bbb C)$ and $ϕ: G_2 → GL_n(\Bbb C)$ are representations. Let $V =M_{mn}(\Bbb C)$ be the vector space of $m×n$-matrices over $\Bbb C$. Define $τ : G → GL(V )$ by $τ(g_1,g_2)(A) = ρ_{g_1}Aϕ_{g_2}^T$ where $B^T$ is the transpose of a matrix $B$.

1. Show that $τ$ is a representation of $G$.
2. Prove that $χ_τ (g_1, g_2) = χ_ρ(g_1)χ_ϕ(g_2)$.
3. Show that if $ρ$ and $ϕ$ are irreducible, then $τ$ is irreducible.
4. Prove that every irreducible representation of $G_1×G_2$ can be obtained in this way.

The solution of 1 is pretty clear.

I am facing problem from 2 onwards. For a representation $\phi : G \to GL(V)$ where $V \cong \Bbb C^n$

$\chi _{\phi}(g)=Tr(\phi_g)=\sum_{i=1}^n<\phi_g(e_i),e_i>$ but what to do here I was writting in terms of $E_{ij}$ bit not getting satisfactory answer. Please help from 2 onwards.

Answer 1

You have the right idea: for 2, we note that the inner product over $M_{mn}$ is given by $\langle A,B \rangle = tr(AB^*)$, with $*$ denoting the conjugate transpose. In other words, our inner product is simply the "dot product" of matrices. Note also that $E_{ij} = e_ie_j^T$. We then have $$ \chi_\tau(g_1,g_2) = \sum_{i= 1}^m \sum_{j=1}^n \langle \tau(g_1,g_2)E_{ij}, E_{ij} \rangle = \sum_{i= 1}^m \sum_{j=1}^n \langle \rho_{g_1}e_ie_j^T\phi_{g_2}^T, e_ie_j^T \rangle = \\ \sum_{i= 1}^m \sum_{j=1}^n \langle \rho_{g_1}e_i e_j^T\phi_{g_2}^T, e_ie_j^T \rangle = \\ \sum_{i= 1}^m \sum_{j=1}^n Tr( \rho_{g_1}e_ie_j^T\phi_{g_2}^Te_je_i^T) = \\ \sum_{i= 1}^m \sum_{j=1}^n Tr( e_i^T\rho_{g_1}e_ie_j^T\phi_{g_2}^Te_j) = \\ \sum_{i= 1}^m \sum_{j=1}^n [e_i^T\rho_{g_1}e_i][e_j^T\phi_{g_2}^Te_j] = \\ \sum_{i= 1}^m \sum_{j=1}^n \langle\rho_{g_1}e_i,e_i\rangle \langle \phi_{g_2} e_j,e_j\rangle = \\ \chi_\rho(g_1) \chi_{\phi}(g_2) $$ For 3: I would try to prove by contrapositive. That is: show that if $\tau$ is reducible but $\phi$ is irreducible, then $\rho$ is reducible. Not sure about 4.