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 $\chi_{\alpha \otimes \beta} = \chi_\alpha\chi_\beta$.