= Schur orthogonality relations
{c}
{title2=$\mathbb E\rho_{ij}\overline{\sigma_{kl}}=d_\rho^{-1}\delta_{\rho\sigma}\delta_{ik}\delta_{jl}$}
For inequivalent chosen <unitary irreducible representations> of a <finite group>, uniform <expectation> gives
$$
\mathbb E_x\rho(x)_{ij}\overline{\sigma(x)_{kl}}
=\begin{cases}d_\rho^{-1}\delta_{ik}\delta_{jl},&\rho=\sigma,\\0,&\rho\ne\sigma.\end{cases}
$$
The <Kronecker deltas> in the first case refer to entries in the same chosen <basis>. Averaging an <intertwiner> and applying the <Schur lemma> proves the formula; summing diagonal entries gives <character orthogonality>.
Back to article page
