# 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 $G$ be a finite group and let $V$ and $W$ be irreducible $G$-modules. Then, every $G$-module homomorphism $f:V\to W$ is either invertible or the trivial zero map.

###### Proof.

Note that both the kernel, $\ker f$, and the image, $\operatorname{im}f$, are $G$-submodules of $V$ and $W$, respectively. Since $V$ is irreducible, $\ker f$ is either trivial or all of $V$. In the former case, $\operatorname{im}f$ 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.

###### Corollary.

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

###### Proof.

Since the ground field is algebraically closed, the linear transformation $f:V\to V$ has an eigenvalue; call it $\lambda$. By definition, $f-\lambda 1$ is not invertible, and hence equal to zero by Schur’s lemma. In other words, $f=\lambda$, a scalar. ∎

Title Schur’s lemma SchursLemma 2013-03-22 13:08:01 2013-03-22 13:08:01 rmilson (146) rmilson (146) 22 rmilson (146) Theorem msc 20C99 msc 20C15 GroupRepresentation DenseRingOfLinearTransformations