Still the problem of giving an orthonormal basis of $L^2 …

However, the connection between them is not always emphasized and the Orthogonality relations are proven more computationally. An immediate outcome of the Schür lemmas is the orthogonality theorem, which most generally states (17a) ∑ g ∈ G Γ j , l U ( g ) ( Γ m , n V ( g - 1 ) ) = n G d U δ U V δ j n δ l m for any pair of irreducible matrix representations Γ U and Γ V of a group G or Schur Orthogonality If $G$ is nonabelian, the group of linear characters is too small to be an orthonormal basis of $L^2 (G)$. Schur’s Lemma and the Schur Orthogonality Relations.
December 21, 2016February 6, 2018/ Asvin G. Both Schur’s Lemma and the Schur Orthogonality relations are part of the basic foundation of representation theory.


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