Schur's Lemma states that any homomorphism between two irreducible representations of a group is either the zero map or an isomorphism, and that any endomorphism of an irreducible representation over an algebraically closed field is a scalar multiple of the identity. Named for Issai Schur, it is a foundational result of representation theory underlying the classification of irreducible representations.
Facts
StatementIn representation theory, Schur's Lemma states that a homomorphism between two finite-dimensional irreducible representations of a group is either invertible or the zero map, and that a linear map commuting with an irreducible representation over an algebraically closed field must be a scalar multiple of the identity. 1 Classification
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Sources
1. Schur's Lemma (Representation Theory) (Wikipedia)
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Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen.
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