 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

For a <p-regular element> $g$, diagonalize its actions on $V$ and $V'$. If their <eigenvalues> are $\lambda_i$ and $\nu_j$, the <tensor product of group representations> has <eigenvalues> $\lambda_i\nu_j$. The lift of <roots of unity> is multiplicative, so
$$
\chi_{V\otimes V'}(g)=\sum_{i,j}\widehat\lambda_i\widehat\nu_j=\chi_V(g)\chi_{V'}(g).
$$
Thus $\boxed{\chi_{V\otimes V'}=\chi_V\chi_{V'}}$ as <Brauer characters>.

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