So I am working on a problem from Steinberg's representation theory of finite groups. I am stuck on exercise $4.11$ (the second part) and can't find the solution anywhere. It goes like this:
Let $G_{1}$ and $G_{2}$ be finite groups and let $G=G_{1}\times G_{2}$. Suppose $\rho:G_{1}\rightarrow GL_{m}(\mathbb{C})$ and $ \varphi:G_{2}\rightarrow GL_{n}(\mathbb{C})$ are representations. Let $V=M_{mn}(\mathbb{C})$ . Define $\tau:G\rightarrow GL(V)$ by $\tau_{(g_{1},g_{2})}(A)=\rho_{g_{1}}A\varphi _{g_{2}}.$
The first part asked to prove it was a representation and that was doable, but the second part i find confusing: prove that $$\chi_{\tau}(g_{1},g_{2})=\chi_{\rho}(g_{1})\chi_{\varphi}(g_{2}). $$ I for one am confused about how I must read $\tau_{(g_{1},g_{2})}$ as a matrix as it's sandwiched in between the two functions so to speak. Do I need to find a matrix such that when multiplied by a matrix $A$, it yields $\tau_{(g_{1},g_{2})}(A)$? Secondly, I am confused on how to find the trace of $\tau_{(g_{1},g_{2})}$ in the first place. Isn't it an $m\times n$ matrix? If not, (again) what form does it have? Thanks in advance for any help.