DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Brauer pair for a block

Definition

A local Brauer pair is $(P, e)$ where $P \leq G$ is a $p$ -subgroup and $e$ is a primitive central idempotent of $k C_{G} (P)$ . For a global block $b \in k G$ , it is a $b$ -Brauer pair when ${Br}_{P} (b) e = e$ . Equivalently the product is nonzero, by Central idempotents under the Brauer homomorphism. Membership in a global block is not an inclusion relation between local pairs. In particular $(1, b)$ is a $b$ -pair because ${Br}_{1}$ is the identity.

Sources

Jacobsen, Block fusion systems and the center of the group ring, §§1.1 and 2.2, pp.3–8 and 13–18. Local argument and conventions as displayed above.

Depends on

Central idempotents under the Brauer homomorphism

Used by

Normal inclusion of Brauer pairs Definition

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

Jacobsen, Block fusion systems and the center of the group ring, §§1.1 and 2.2, pp.3–8 and 13–18 (standard reference, not scraped)