LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-05

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The Brauer character is independent of basis and splitting-field realization

Statement

For a finite-dimensional $k G$ -module $V$ , the value $φ_{V} (g)$ on a $p$ -regular element $g$ is independent of the chosen basis of $V$ . It is also independent of the chosen splitting realization, provided the realizations identify the same prime-to- $p$ roots of unity by the residue-field isomorphism.

Facts & Assumptions

Given: A finite-dimensional $k G$ -module $V$ and a $p$ -regular element $g \in G$ .

[F1]

The Brauer character of $V$ at $g$ is the sum of the Teichmuller lifts of the eigenvalues of the action of $g$ on $V$ (Brauer character of a finite-dimensional kG-module).

[L1]

Teichmuller lifts are unique and multiplicative (The Teichmuller lift is multiplicative and unique).

Proof

technique · direct

1.1

Changing basis replaces the matrix of $g$ by a similar matrix. Similar matrices have the same characteristic polynomial, hence the same eigenvalues with the same multiplicities. Therefore the sum in [F1] is basis-independent.

F1givenalgebra

1.2

If two splitting realizations identify the same prime-to- $p$ roots of unity in the residue fields, then they identify each eigenvalue of the action of $g$ and, by [L1], identify its unique Teichmuller lift. So the lifted eigenvalue sum computed in [F1] is the same in either realization.

F1L1algebra

2.1

Steps 1.1 and 1.2 prove both independence statements.

step 1.1step 1.2∎

Depends on

Used by

Brauer characters are class functions on p-regular elements Proposition
Brauer characters are additive on short exact sequences Theorem

Dependency tree · two levels

4 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)