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

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Brauer characters are additive on short exact sequences

Statement

For every short exact sequence of finite-dimensional $k G$ -modules

$0 \to U \to V \to W \to 0,$

one has

$φ_{V} = φ_{U} + φ_{W}$

on $G^{0}$ . In particular Brauer characters are additive on direct sums.

Facts & Assumptions

Given: A short exact sequence $0 \to U \to V \to W \to 0$ of finite-dimensional $k G$ -modules.

[F1]

The Brauer character at a $p$ -regular element is the lifted sum of the eigenvalues of that element on the module (Brauer character of a finite-dimensional kG-module).

[L1]

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

Proof

technique · direct

1.1

Fix $g \in G^{0}$ . Choose a basis of $V$ whose first block is a basis of the $g$ -stable submodule $U$ . In that basis the action matrix of $g$ on $V$ has block upper-triangular form $(\begin{array}{cc} A & * \\ 0 & B \end{array}),$ where $A$ is the action on $U$ and $B$ is the induced action on $W$ .

givenchoosealgebra

2.1

A block upper-triangular matrix has eigenvalue multiset equal to the union of the eigenvalue multisets of its diagonal blocks. Therefore the eigenvalues of $g$ on $V$ are exactly those on $U$ together with those on $W$ . Using [F1] and [L1], the lifted traces add: $φ_{V} (g) = φ_{U} (g) + φ_{W} (g)$ .

F1L1step 1.1algebra

3.1

Since $g \in G^{0}$ was arbitrary, $φ_{V} = φ_{U} + φ_{W}$ on $G^{0}$ . The direct-sum case is the split short exact sequence.

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)