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Schur's Lemma

  • Definition: a) Let p and p' be irreducible representations of G on vector spaces V and V', respectively, and let T:V' -> V be a G-invariant transformation. Either T is an isomorphism, or else T=0. b) Let p be an irreducible representation of G on a vector space V, and let T:V -> V be a G-invariant linear operator. Then T is multiplication by a scalar: T=cI.

    Source: Algebra, second edition by Michael Artin