TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Brauer-Nesbitt determines semisimplifications

Statement

Let $V$ and $W$ be finite-dimensional $k G$ -modules over a splitting field of characteristic $p$ . Then

$φ_{V} = φ_{W}$

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

Facts & Assumptions

Given: 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

1.1

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 $W$ .

L1givenalgebra

2.1

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 $φ_{V} = φ_{W}$ .

step 1.1

2.2

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 $V^{s s} ≅ W^{s s}$ .

L2step 1.1algebra

3.1

Steps 2.1 and 2.2 prove the equivalence.

step 2.1step 2.2∎

Depends on

Used by

FALSE: modular representations are determined by ordinary characters False statement

Dependency tree · two levels

8 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

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