Solution
= Solution
For representation spaces $V,W$, the <tensor product of group representations> acts on product basis vectors through
$$
D_{V\otimes W}(g)=D_V(g)\otimes D_W(g).
$$
Its <character> is the <trace>. Summing diagonal elements explicitly,
$$
\chi_{V\otimes W}(g)=\sum_{i,j}[D_V(g)]_{ii}[D_W(g)]_{jj}
=\boxed{\chi_V(g)\chi_W(g)}.
$$
No step assumes one-dimensional matrices, so the result applies to both degenerate and nondegenerate representations.