CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Schur's lemma for irreducible representations: a nonzero intertwiner is an isomorphism, and ${End}_{G} (V)$ is a division ring

Statement

Let $V$ and $W$ be irreducible representations of a group $G$ over a field $k$ . Then every nonzero intertwiner $f : V \to W$ is an isomorphism. Consequently ${End}_{G} (V)$ is a division ring.

Facts & Assumptions

Given: Irreducible representations $V$ and $W$ of $G$ over a field $k$ .

[L1]

Irreducible representations are exactly simple $k [G]$ -modules (Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules).

[L2]

$G$ -equivariant maps are exactly $k [G]$ -module homomorphisms (For a commutative ring $R$ , $R$ -linear $G$ -actions are exactly the compatible left $R [G]$ -module structures).

[L3]

A nonzero homomorphism between simple modules is an isomorphism, and the endomorphism ring of a simple module is a division ring (Schur's lemma for simple modules).

Proof

technique · direct

1.1

By [L1], the irreducible representations $V$ and $W$ are simple $k [G]$ -modules, and by [L2] a nonzero intertwiner $f : V \to W$ is a nonzero $k [G]$ -module homomorphism between them.

L1L2given

2.1

Applying [L3] to that module homomorphism shows that $f$ is an isomorphism. Applying [L3] with $V = W$ shows that ${End}_{G} (V)$ is a division ring.

step 1.1L3∎

Depends on

Used by

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory, Theorem 2.1.1 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Proposition 1.16 (standard reference, not scraped)

Schur's lemma for irreducible representations: a nonzero intertwiner is an isomorphism, and End⁡G(V)EndG​(V) is a division ring