Schur’s lemma
Schur’s lemma is a fundamental result in representation theory,
an elementary observation about irreducible modules, which is nonetheless
noteworthy because of its profound applications.
Lemma (Schur’s lemma).
Let be a finite group and let and be irreducible
-modules. Then, every -module homomorphism
is
either invertible
or the trivial zero map
.
Proof.
Note that both the kernel, , and the image, , are -submodules of and
, respectively. Since is irreducible, is either
trivial or all of . In the former case, is all of
— also because is irreducible — and hence is invertible. In
the latter case, is the zero map.
∎
One of the most important consequences of Schur’s lemma is the following.
Corollary.
Let be a finite-dimensional, irreducible -module taken over an algebraically closed field. Then, every -module homomorphism is equal to a scalar multiplication.
Proof.
Since the ground field is algebraically closed, the linear
transformation has an eigenvalue
; call it .
By definition, is not invertible, and hence equal to
zero by Schur’s lemma. In other words, , 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 |