DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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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p-blocks from primitive central idempotents

Definition

Fix a splitting $p$ -modular system $(K, O, k)$ for a finite group $G$ . A $p$ -block idempotent is a primitive central idempotent of $O G$ or, after reduction, of $k G$ .

If $e$ is such an idempotent, the corresponding block is the two-sided ideal

$B = O G e or B = k G e .$

The primitive central idempotents are pairwise orthogonal and sum to $1$ , so the algebra splits as a direct product of its block summands.

Depends on

A splitting p-modular system for a finite group is a p-modular system whose fraction and residue fields split the needed group algebras

Used by

Every block contains a module whose vertex is a full defect group Corollary
A block induced from a subgroup Definition
Block bimodule for the double group Definition
Brauer homomorphism for a p subgroup Definition
Generalized Jucys-Murphy primary idempotents and the residue-summand functors Eᵢ, Fᵢ Definition
A block with one ordinary and one Brauer character Example
Block idempotents lift uniquely from kH to OH Lemma
Central idempotents under the Brauer homomorphism Proposition
Weight-zero symmetric-group blocks are defect zero Proposition
Blocks partition the ordinary and Brauer irreducible characters Theorem
Modular block central characters correspond to blocks Theorem
Nakayama block classification for symmetric groups 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)