= 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 $(g,h)\mapsto\chi_V(g)\chi_W(h)$. Over a splitting field 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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