PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Central idempotents under the Brauer homomorphism

Statement

For a central block idempotent $b \in k G$ , ${Br}_{P} (b)$ is a central idempotent of $k C_{G} (P)$ , possibly zero. It equals the sum of exactly those primitive central idempotents $e$ for which ${Br}_{P} (b) e = e$ .

Facts & Assumptions

Given: A block $b$ and a $p$ -subgroup $P$ .

[F1]

The Brauer map is a unital surjective algebra homomorphism. (Brauer homomorphism is multiplicative)

[F2]

The identity is the sum of orthogonal primitive central block idempotents. (p-blocks from primitive central idempotents)

Proof

technique · direct

1.1F1

Multiplicativity gives ${Br}_{P} (b)^{2} = {Br}_{P} (b)$ . Every element $c$ of the target has a lift $a$ in the domain. Since $b$ commutes with $a$ , their images commute, so ${Br}_{P} (b)$ is central.

2.1F2step 1.1∎

Write $1 = \sum e_{j}$ for the block decomposition of the centralizer algebra. Each ${Br}_{P} (b) e_{j}$ is a central idempotent below primitive $e_{j}$ , so is either zero or $e_{j}$ : otherwise it and $e_{j} - {Br}_{P} (b) e_{j}$ split $e_{j}$ . Multiplying the decomposition of one by ${Br}_{P} (b)$ yields exactly the asserted sum. If all products vanish, the sum is empty and equals zero.

Sources

Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245. Local argument and conventions as displayed above.

Depends on

Used by

Brauer pair for a block Definition
Centralizer containment makes block induction well-defined Lemma

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

Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245 (standard reference, not scraped)