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Schur's lemma

Two related basic facts in representation theory bear the name of Schur: one general about categories of modules, and another specific to the case of complex numbers. If $G$ is a group, a $G$-module is any $k$-module with an action of $G$, where $k$ is a fixed commutative unital ground ring.

1. Given a group $G$ and a linear map $\varphi :M\to N$ between two irreducible (= simple) $G$-modules (linear maps intertwining the actions) then $\varphi$ is either the zero morphism or an isomorphism. It follows that the endomorphisms of simple irreducible $G$-module form a division ring.

2. Set $M=N$, and suppose further that the ground ring $k$ is an algebraically closed field; then $\varphi$ is a multiple $\lambda I$ of the identity operator. In other words, the nontrivial automorphisms of simple modules, a priori possible by (1), are ruled out over algebraically closed fields.

Part (1) is essentially category-theoretic and can be generalized in many ways, for example, by replacing $G$ by some $k$-algebra and taking the representations compatible with the action of $k$; more generally, given an abelian category, the endomorphism ring of a simple object is a division ring. Schur’s lemma is one of the basic facts of representation theory. For (2), if the endomorphism rings of all objects in an abelian category are finite-dimensional over an algebraically closed ground field $k$ (as is the case for group representations), then the endomorphism ring of a simple object is $k$ itself.