Schur's Lemma

Equivalently, $f$ is a module homomorphism for $kG$-modules, where $kG$ is the group algebra.

Let $V,W$ be irreducible representations of $G$ over $k$. Then:

1. Every nonzero $G$-intertwiner $f:V\to W$ is an isomorphism. In particular, if $V\not\cong W$, then $\mathrm{Hom}_G(V,W)=0$.

1. The endomorphism ring $\mathrm{End}_G(V)=\mathrm{Hom}_G(V,V)$ is a division ring (every nonzero element is invertible).

1. If $k$ is algebraically closed and $V$ is finite-dimensional (e.g. $k=\mathbb{C}$), then

i.e. every $G$-equivariant endomorphism is scalar multiplication.

This result is frequently paired with character theory and the corollary that central elements act by scalars on irreducibles (see Schur's corollary).

Example 1: Cyclic group $C_n$ over $\mathbb{C}$

All irreducible $\mathbb{C}$-representations of $C_n$ are 1-dimensional. If $\chi,\psi$ are distinct 1-dimensional characters, then any $C_n$-equivariant map $f:\mathbb{C}_\chi\to \mathbb{C}_\psi$ must be zero, because equivariance forces

for all $g$, and choosing $g$ with $\chi(g)\neq \psi(g)$ implies $f(v)=0$. If $\chi=\psi$, then $\mathrm{End}_{C_n}(\mathbb{C}_\chi)=\mathbb{C}$.

Example 2: The standard 2D irreducible of $S_3$

Let $V$ be the standard 2-dimensional irreducible representation of $S_3$ (e.g. the reflection representation on the plane of an equilateral triangle). Schur’s lemma over $\mathbb{C}$ says any $S_3$-equivariant $T:V\to V$ must be of the form $T=\lambda I$. So the commutant $\{T\in \mathrm{End}(V): T\rho(g)=\rho(g)T \ \forall g\}$ is exactly $\mathbb{C}\cdot I$.

Example 3: A real irreducible where $\mathrm{End}_G(V)\neq \mathbb{R}$

Let $G=C_3=\langle r\rangle$ act on $V=\mathbb{R}^2$ by rotation by $120^\circ$. This is an irreducible real representation. The real matrices commuting with a $120^\circ$ rotation are precisely the real linear combinations of $I$ and that rotation (equivalently, they identify with $\mathbb{C}$ acting on $\mathbb{R}^2\cong \mathbb{C}$). Thus

illustrating the “division ring” conclusion over non-algebraically-closed fields.