Schur’s lemma

Schur’s lemma is a fundamental result in representation theory, an elementary observation about irreducible modulesMathworldPlanetmath, which is nonetheless noteworthy because of its profound applications.

Lemma (Schur’s lemma).

Let G be a finite groupMathworldPlanetmath and let V and W be irreduciblePlanetmathPlanetmath G-modules. Then, every G-module homomorphismMathworldPlanetmath f:VW is either invertiblePlanetmathPlanetmathPlanetmath or the trivial zero mapMathworldPlanetmath.


Note that both the kernel, kerf, and the image, imf, are G-submodulesMathworldPlanetmath of V and W, respectively. Since V is irreducible, kerf is either trivial or all of V. In the former case, imf is all of W — also because W is irreducible — and hence f is invertible. In the latter case, f is the zero map. ∎

One of the most important consequences of Schur’s lemma is the following.


Let V be a finite-dimensional, irreducible G-module taken over an algebraically closed field. Then, every G-module homomorphism f:VV is equal to a scalar multiplication.


Since the ground field is algebraically closedMathworldPlanetmath, the linear transformation f:VV has an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath; call it λ. By definition, f-λ1 is not invertible, and hence equal to zero by Schur’s lemma. In other words, f=λ, a scalar. ∎

Title Schur’s lemma
Canonical name SchursLemma
Date of creation 2013-03-22 13:08:01
Last modified on 2013-03-22 13:08:01
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 22
Author rmilson (146)
Entry type Theorem
Classification msc 20C99
Classification msc 20C15
Related topic GroupRepresentation
Related topic DenseRingOfLinearTransformations