Theorem

Schur's Lemma

Representation Theory · example.tex

Let $(\rho, V)$ and $(\sigma, W)$ be irreducible representations (Irreducible Representation) of $G$ over an algebraically closed field $k$.

Any $G$-module homomorphism $\phi : V \to W$ is either zero or an isomorphism.
$\mathrm{End}_G(V) \cong k$.

Proof

(1) Since $\ker(\phi)$ and $\mathrm{Im}(\phi)$ are subrepresentations of $V$ and $W$ respectively, irreducibility forces each to be $\{0\}$ or the whole space. The non-trivial cases give an isomorphism. (2) For any $\lambda \in k$, the map $\phi - \lambda \cdot \mathrm{id}_V$ is again a $G$-endomorphism. Since $k$ is algebraically closed, $\phi$ has some eigenvalue $\lambda_0$, and $\phi - \lambda_0 \cdot \mathrm{id}_V$ has a nontrivial kernel. By (1) it must be identically zero, so $\phi = \lambda_0 \cdot \mathrm{id}_V$.

Depends on

Irreducible Representation

Dependency Graph

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