# Proof of the second part of Schur's lemma

## Background

According to PlanetMath Schur's lemma is this:

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.

### 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.

## Question

In the proof above, why can Schur's lemma be applied to $\,f - \lambda 1$? Is it assumed that $\,f - \lambda 1$ is also a $G$-module homomorphism?

• The set of all $G$-module homomorphisms from $V$ to $W$ is a vector space, and it contains $1$. – Qiaochu Yuan Sep 10 '12 at 22:47

$f$ is a $G$-module homomorphisms by assumption. Let $\rho : G \rightarrow GL_k(V)$ by the representation, where $k$ is the field. Then you have
$(f - \lambda I)(\rho(g)(v)) = f(\rho(g)(v)) - \lambda I(\rho(g)( v)) = \rho(g)(f(v)) - \lambda \rho(g)(v)$
by using the fact that $f$ is a $G$-module homomorphism. Since $\rho(g) \in GL_k(V)$, this means that $\rho(g)$ is a k-linear transformation, hence
$= \rho(g)(f(v)) - \rho(g)(\lambda v) = \rho(g)(f(v) - \lambda I(v)) = \rho(g)((f - \lambda I))(v)$
So $f - \lambda I$ has been shown to be $G$-linear.