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Concept
Schur's lemma
Summary
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations
of a group G and φ is a linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0. An important special case occurs when M = N, i.e. φ is a self-map; in particular, any element of the center of a group must act as a scalar operator (a scalar multiple of the identity) on M. 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. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen.
Representation theory is the study of homomorphisms from a group, G, into the general linear group GL(V) of a vector space V; i.e., into the group of automorphisms of V. (Let us here restrict ourselves to the case when the underlying field of V is , the field of complex numbers.) Such a homomorphism is called a representation of G on V. A representation on V is a special case of a group action on V, but rather than permit any arbitrary bijections (permutations) of the underlying set of V, we restrict ourselves to invertible linear transformations.
Let ρ be a representation of G on V. It may be the case that V has a subspace, W, such that for every element g of G, the invertible linear map ρ(g) preserves or fixes W, so that (ρ(g))(w) is in W for every w in W, and (ρ(g))(v) is not in W for any v not in W. In other words, every linear map ρ(g): V→V is also an automorphism of W, ρ(g): W→W, when its domain is restricted to W. We say W is stable under G, or stable under the action of G. It is clear that if we consider W on its own as a vector space, then there is an obvious representation of G on W—the representation we get by restricting each map ρ(g) to W.
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