TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-09-07

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Absolute irreducibility via the endomorphism division algebra

Statement

Let $F$ have characteristic $0$ , $G$ be finite, and $V$ be an irreducible finite-dimensional $F$ -representation. Then $V$ is absolutely irreducible if and only if $D_{V} = F$ (via scalar endomorphisms).

Facts & Assumptions

Given: $F$ , $G$ , and $V$ as in the statement.

[L1]

Base change gives $E \otimes_{F} D_{V} ≅ {End}_{G} (E \otimes_{F} V)$ for every field extension $E / F$ (Base change for intertwiner spaces).

[L2]

Over a finite splitting field, scalar extension of $V$ is a common multiple of one Galois orbit of absolutely irreducible constituents (Scalar extension of an irreducible finite-group representation).

Proof

technique · direct

1.1

Choose a finite splitting field $E / F$ . If $V$ is absolutely irreducible, then $E \otimes_{F} V$ is irreducible, so its endomorphism algebra is $E$ . By [L1], $E \otimes_{F} D_{V} ≅ E$ , and comparing $F$ -dimensions gives $D_{V} = F$ .

L1choose

1.2

Conversely assume $D_{V} = F$ . Then [L1] makes ${End}_{G} (E \otimes_{F} V)$ one-dimensional over $E$ .

L1given

2.1

In the decomposition of [L2], either a multiplicity exceeds one or two inequivalent constituents occur whenever $E \otimes_{F} V$ is reducible; either case supplies a non-scalar projection endomorphism. This contradicts step 1.2, so $E \otimes_{F} V$ is irreducible and $V$ is absolutely irreducible.

L2step 1.2∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Gabor Wiese, Galois Representations, Theorem 2.3.11 (standard reference, not scraped)
Weizhe Zheng, Lectures on Algebra, Theorem 4.2.3 (standard reference, not scraped)