Outer tensor product establishes bijection between irreducible representations of direct factors and direct product

Statement

Suppose $G$ and $H$ are groups and $k$ is a field. Let $\operatorname{Irr}(S)$ denote the set of irreducible representation of the group $S$ over $k$. Then, there is a natural bijection:

$\operatorname{Irr}(G) \times \operatorname{Irr}(H) \leftrightarrow \operatorname{Irr}(G \times H)$.

The bijection is given using the outer tensor product of linear representations, as follows. For irreducible representations $\alpha,\beta$ of $G$ and $H$, to vector spaces $V$ and $W$, $\alpha \otimes \beta$ is defined as a linear representation on the tensor product $V \otimes W$, with:

$(\alpha \otimes \beta)(g,h) = \alpha(g) \otimes \beta(h)$.