Schur’s lemma

Schur’s lemma is a fundamental result in representation theory,an elementary observation about irreducible modules, which is nonethelessnoteworthy 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 $f\mathrm{:}V\mathrm{\to }W$ iseither invertible or the trivial zero map.

Proof.

Note that both the kernel, $\ker f$, and the image, $\mathrm{im}f$, are $G$-submodules of $V$ and$W$, respectively. Since $V$ is irreducible, $\ker f$ is eithertrivial or all of $V$. In the former case, $\mathrm{im}f$ is all of $W$— also because $W$ is irreducible — and hence $f$ is invertible. Inthe 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 overan algebraically closed field. Then, every $G$-module homomorphism$f\mathrm{:}V\mathrm{\to }V$ is equal to a scalar multiplication.

Proof.

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