DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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Brauer character of a finite-dimensional kG-module

Definition

Fix a splitting $p$ -modular system $(K, O, k)$ for a finite group $G$ , and let $V$ be a finite-dimensional $k G$ -module. For $g \in G^{0}$ , the element $g$ has order prime to $p$ , so any matrix representing its action on $V$ satisfies $X (g)^{∣ g ∣} = I$ with separable polynomial $t^{∣ g ∣} - 1$ . Hence $X (g)$ is diagonalizable over $k$ , with eigenvalues $λ_{1}, \dots, λ_{n}$ of order prime to $p$ .

The Brauer character of $V$ is the function

$φ_{V} : G^{0} \to K, φ_{V} (g) : = τ (λ_{1}) + \dots + τ (λ_{n}),$

where $τ$ denotes the Teichmuller lift.

So the Brauer character is the lifted trace of the action of $g$ on $V$ , but only on the $p$ -regular part of $G$ .

Depends on

Used by

Ordinary trace on a p-singular unipotent element is not a Brauer-character value Counterexample
Brauer characters of a p-group Example
FALSE: a Brauer character is defined on all elements by the usual trace False statement
The Brauer character is independent of basis and splitting-field realization Lemma
Brauer characters are class functions on p-regular elements Proposition
Brauer characters are additive on short exact sequences Theorem

Dependency tree · two levels

3 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)