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

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Brauer characters are class functions on p-regular elements

Statement

If $V$ is a finite-dimensional $k G$ -module, then its Brauer character $φ_{V}$ is constant on $p$ -regular conjugacy classes.

Facts & Assumptions

Given: A finite-dimensional $k G$ -module $V$ and $p$ -regular elements $x, y \in G$ with $y = g x g^{- 1}$ .

[F1]

The Brauer character at a $p$ -regular element is computed from the eigenvalues of the action matrix (Brauer character of a finite-dimensional kG-module).

[L1]

That value is independent of the chosen basis (The Brauer character is independent of basis and splitting-field realization).

Proof

technique · direct

1.1

In any basis of $V$ , the matrices representing $x$ and $y = g x g^{- 1}$ are similar, because the representation map sends conjugation in $G$ to conjugation in $GL (V)$ . So they have the same eigenvalues with the same multiplicities.

givenalgebra

2.1

By [F1], the Brauer character is the sum of the Teichmuller lifts of those eigenvalues, and [L1] shows that the value does not depend on which basis was used to see the similarity. Hence $φ_{V} (x) = φ_{V} (y)$ .

F1L1step 1.1

3.1

Therefore $φ_{V}$ is a class function on $G^{0}$ .

step 2.1∎

Depends on

Used by

Dependency tree · two levels

5 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)