TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-26 (gpt-6-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Brauer-Nesbitt determines semisimplifications

Statement

Fix a finite group $G$ , a prime $p$ , and a splitting $p$ -modular system $(K, O, k)$ for $G$ (A splitting p-modular system for a finite group is a p-modular system whose fraction and residue fields split the needed group algebras). Let $V$ and $W$ be finite-dimensional $k G$ -modules, with Brauer characters defined using this system (Brauer character of a finite-dimensional kG-module). Then

$φ_{V} = φ_{W}$

if and only if the semisimplifications $V^{s s}$ and $W^{s s}$ are isomorphic.

Facts & Assumptions

Given: A finite group $G$ , a prime $p$ , a fixed splitting $p$ -modular system $(K, O, k)$ for $G$ , and finite-dimensional $k G$ -modules $V$ and $W$ .

[L1]

Brauer characters are additive on short exact sequences (Brauer characters are additive on short exact sequences).

[L2]

The irreducible Brauer characters form a basis of the complex vector space of class functions on $G^{0}$ (Irreducible Brauer characters form a basis of the p-regular class functions).

Proof

technique · iff

1.1L1givenalgebra

Repeatedly applying [L1] along a composition series of $V$ shows that $φ_{V}$ is the sum of the irreducible Brauer characters of the composition factors of $V$ , counted with multiplicity. The same holds for $W$ .

2.1step 1.1

If $V^{s s} ≅ W^{s s}$ , then $V$ and $W$ have the same irreducible composition factors with the same multiplicities. Step 1.1 then gives $φ_{V} = φ_{W}$ .

2.2L2step 1.1algebra

Conversely, suppose $φ_{V} = φ_{W}$ . Subtracting the two expansions from step 1.1 gives a linear relation among irreducible Brauer characters. By [L2], those characters are linearly independent, so every coefficient is $0$ . Hence the composition multiplicities in $V$ and $W$ agree, and therefore $V^{s s} ≅ W^{s s}$ .

3.1step 2.1step 2.2∎

Steps 2.1 and 2.2 prove the equivalence.

Depends on

Used by

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Dependency tree · two levels

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Sources

J. Miquel Martinez, Modular Representation Theory of Finite Groups (standard reference, not scraped)
Tudor Ciurca, Representation Theory (standard reference, not scraped)