= Schur averaging of rectangular matrices
{c}
{title2=$\mathbb E_x\rho(x)M\sigma(x)^*$}
For chosen inequivalent <unitary irreducible representations> $\rho,\sigma$ of a <finite group> and a $d_\rho\times d_\sigma$ matrix $M$, the average $\mathbb E_x\rho(x)M\sigma(x)^*$ is zero when $\rho\ne\sigma$. For $\rho=\sigma$, it is $(\operatorname{tr}M/d_\rho)I$. This follows from the <Schur lemma>, since the average intertwines the two <group representations>. Equivalent representations in different bases require the corresponding <intertwiner>; the identity-matrix formula assumes literally the same representative.
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