Schur's lemma

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In mathematics, Schur's lemma is now a generic term applied to theorems on the commutant of a module M that is simple. It is named for Issai Schur.

The original case may have been for linear representations of a finite group G over the complex number field C (it does not apply to real fields). If G acts irreducibly on a finite-dimensional complex vector space V through a group representation ρ, then the only linear transformations α of V to itself, such that

αρ(g) = ρ(g)α

for all g in G, are scalar multiples of the identity transformation. Here an irreducible representation on V is simply one with no invariant subspaces aside from {0} and V itself (because the group algebra is semisimple).

Since scalar multiples of the identity transformation trivially commute with all linear transformations, one can say that the import of the lemma is in this case that the commutant of the representation is as small as possible. The condition of irreducibility is necessary because a non-trivial invariant subspace would be the image of a projection operator that would commute with the ρ(g) (see Maschke's theorem).

There are many generalisations: to other fields, to Lie groups and Lie algebras, and in module theory. In the latter, a theorem commonly called Schur's Lemma states that the endomorphism ring End_RM of any simple R-module M is a division ring.

[edit] Alternative formulation

For those readers who do not know what the commutant of a simple module is, Schur's lemma may be formulated as follows: Let D(g) be a unitary, irreducible matrix representation of a group G = {g}. The matrices D(g) are of dimension f x f. If A is a matrix of dimension f x f commuting with all D(g),

then A = c E , where c is a scalar (a complex number) and E is the f x f unit matrix. Conversely, if the only matrix that commutes with all unitary matrices D(g), representing G, is equal to a scalar times the unit matrix, then the matrix representation D(g) of G is irreducible.

[edit] Reference

• David S. Dummit, Richard M. Foote. Abstract Algebra. 2nd ed., pg. 337.