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.
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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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Source Schur's Lemma (Representation Theory) (Wikipedia)
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1. Schur's Lemma (Representation Theory) (Wikipedia)
Wikimedia FoundationLead section, closing sentence
Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen.
In Group: Lie Algebras and Representation Theory, Article text
In mathematics, Schur's lemma is an elementary but useful statement in representation theory of groups and algebras.
Proved By: Issai Schur, History
The lemma is named after Issai Schur who used it to prove the Schur orthogonality relations and develop the basics of the representation theory of finite groups.
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