The Schur orthogonality relations are a set of mathematical statements that arise in the context of representation theory, particularly concerning the representations of the symmetric group and the general linear group. These relations provide a way to understand how different irreducible representations (irreps) of a group are related to one another through their characters.
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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.