ExampleConstruction: Literature-sourcedVerification: 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.

A block with one ordinary and one Brauer character

Example

In the characteristic- $2$ decomposition matrix of $S_{3}$ , the block containing the standard ordinary character has exactly one ordinary irreducible character and one irreducible Brauer character.

Facts & Assumptions

Given: The characteristic- $2$ decomposition matrix of $S_{3}$ , already ordered by blocks, $(\begin{array}{cc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array}),$ whose last row is the ordinary standard character and whose last column is the $2$ -dimensional irreducible Brauer character.

[L1]

A $p$ -group example shows what a one-Brauer-character block looks like (Brauer characters of a p-group).

Verification

technique · direct

1.1

In the displayed block-ordered matrix, the last diagonal sector is the $1 \times 1$ block $(1)$ .

givenalgebra

2.1

That sector therefore corresponds to a block containing exactly one ordinary irreducible character and exactly one irreducible Brauer character, namely the standard row and the $2$ -dimensional Brauer column. This contrasts with [L1], where having one Brauer character does not force a unique ordinary one.

L1step 1.1

3.1

So such one-row one-column blocks do occur.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)