= External tensor product of group representations
{title2=$V\boxtimes W$}
For a $G$-<representation> $V$ and an $H$-<representation> $W$ over the same <field>, $V\boxtimes W$ is the <representation> of $G\times H$ on $V\otimes W$ given by $(g,h)(v\otimes w)=gv\otimes hw$. Its <character> is $(g,h)\mapsto\chi_V(g)\chi_W(h)$. For <finite groups> $G,H$, over a <splitting field for finite group representations> in <characteristic zero> the external products of irreducible <representations> form all <irreducible representations> of $G\times H$: character orthogonality gives irreducibility, and their squared degrees sum to $|G||H|$, proving exhaustion. Unlike the <tensor product> of two representations of one group, the two factors here act independently.
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