Matrix coefficient ($x\mapsto \rho (x{)}_{ij}$)

A matrix coefficient of a finite-dimensional group representation is a scalar function obtained by applying a linear functional to the translate of a fixed vector. In a chosen basis these functions include $x\mapsto \rho (x{)}_{ij}$. For a finite group, the Schur orthogonality relations make the scaled coefficients of all inequivalent unitary irreducible representations an orthonormal basis of its scalar functions.

For inequivalent chosen unitary irreducible representations of a finite group, uniform expectation givesThe 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.

For chosen inequivalent unitary irreducible representations $\rho ,\sigma$ of a finite group and a $d_{\rho }\times d_{\sigma }$ matrix $M$, the average ${\mathbb{E}}_x\rho (x)M\sigma (x{)}^{\ast }$ is zero when $\rho \ne \sigma$. For $\rho =\sigma$, it is $(\mathrm{tr}M/d_{\rho })I$. 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.