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

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.

After block ordering, the decomposition matrix is block diagonal

Statement

If the ordinary irreducible characters and irreducible Brauer characters are ordered block by block, then the decomposition matrix becomes block diagonal.

Facts & Assumptions

Given: The decomposition matrix $D = (d_{χ φ})$ .

[F1]

The entry $d_{χ φ}$ records the multiplicity of the simple $k G$ -module $S_{φ}$ in the reduction of a stable lattice affording $χ$ (Decomposition numbers and the decomposition matrix).

[L1]

Ordinary irreducibles and Brauer irreducibles each belong to unique blocks (Blocks partition the ordinary and Brauer irreducible characters).

Proof

technique · direct

1.1

If $d_{χ φ} \neq 0$ , then by [F1] the simple module $S_{φ}$ occurs in the reduction of a stable lattice affording $χ$ . A composition factor of a module lies in the same block as the module itself, so $χ$ and $φ$ must belong to the same block.

F1L1givenalgebra

2.1

Therefore entries connecting two different blocks are zero. After reordering rows and columns by blocks, all nonzero entries lie inside the matching block sectors, so the matrix is block diagonal.

step 1.1algebra∎

Depends on

Used by

FALSE: every block has one ordinary and one Brauer irreducible character False statement

Dependency tree · two levels

6 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)