# pointwise product of two characters of $G$ is a character of $G$

Let $\phi: G \to GL_{n}(\mathbb{C})$ and $\rho: G \to GL_{m}(\mathbb{C})$ be representations. Let $V = M_{mn}(\mathbb{C})$. Define the representations $\tau : G \to GL(V)$ by $\tau_{g}(A) = \rho_{g}A\phi_{g}^{T}$.

I know that $\chi_{\tau}(g) = \chi_{\rho}(g)\chi_{\psi}(g) \ \forall \ g \in G$. What is the best argument to show that the pointwise product of two characters of $G$ is a character of $G$?

• Hint: What is the character of a tensor product of representations? You've pretty much done it already – leibnewtz Oct 4 '17 at 0:52
• Why do we have $\chi_{\tau}(g) = \chi_{\rho}(g)\chi_{\psi}(g) \ \forall \ g \in G$? – user160919 Mar 12 at 13:01

Have you learned about the tensor product of two representations? The character of $\phi\otimes\rho$ is the pointwise product of the characters of $\phi$ and $\rho$.