Character of tensor product of linear representations is product of characters

Statement

Statement for a field

Suppose $G$ is a group (not necessarily finite) and $\alpha ,\beta$ are finite-dimensional linear representations of $G$ over a field $K$. Denote by $\chi _{\alpha },\chi _{\beta }$ respectively the characters of $\alpha$ and $\beta$. Denote by $\alpha \otimes \beta$ the tensor product of linear representations $\alpha$ and $\beta$ and by $\chi _{\alpha \otimes \beta }$ its character. Then, for any $g\in G$, we have:

$\chi _{\alpha \otimes \beta }(g)=\chi _{\alpha }(g)\chi _{\beta }(g)$

In point-free notation, this states that $\chi _{\alpha \otimes \beta }=\chi _{\alpha }\chi _{\beta }$.