TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-05

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Blocks partition the ordinary and Brauer irreducible characters

Statement

Every ordinary irreducible character and every irreducible Brauer character lies in exactly one block.

Facts & Assumptions

Given: The primitive central idempotents of the modular system of $G$ .

[F1]

A block is the direct-product summand cut out by a primitive central idempotent (p-blocks from primitive central idempotents).

[L1]

Irreducible Brauer characters are attached to simple modules (Irreducible Brauer characters form a basis of the p-regular class functions).

Proof

technique · direct

1.1

Because the primitive central idempotents are central, pairwise orthogonal, and sum to $1$ , every module $M$ decomposes as the direct sum of the eigenspaces $e M$ .

F1givenalgebra

2.1

If $M$ is simple, only one summand in step 1.1 can be nonzero; otherwise $M$ would split as a nontrivial direct sum of submodules. So each simple $k G$ -module, hence each irreducible Brauer character by [L1], lies in exactly one block.

L1step 1.1algebra

3.1

The same argument over characteristic $0$ applies to simple $K G$ -modules and therefore to ordinary irreducible characters. Hence both ordinary and Brauer irreducibles are partitioned by blocks.

F1step 1.1algebra∎

Depends on

Used by

A block with one ordinary and one Brauer character Example
FALSE: every block has one ordinary and one Brauer irreducible character False statement
After block ordering, the decomposition matrix is block diagonal Proposition
Defect groups and Brauer's main theorems lie beyond this page Remark

Dependency tree · two levels

8 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)