CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Generalized decomposition columns have corresponding block support

Statement

Assume the Axiom of Choice. Let $(K, O, k)$ be a splitting $p$ -modular system for a finite group $G$ , with $k$ algebraically closed. Fix a $p$ -element $u$ , put $H = C_{G} (u)$ , let $c$ be a block of $k H$ , and let $φ \in IBr (H, c)$ . If $χ \in {Irr}_{K} (G, B)$ for a block $B$ of $k G$ , then $d_{χ, φ}^{u} \neq 0 ⟹ B = c^{G} .$ Thus the generalized-decomposition column indexed by $φ$ has nonzero rows in at most the single global block $c^{G}$ ; in particular it cannot have nonzero entries in two distinct global blocks.

Facts & Assumptions

Given: AC and the modular system, element, centralizer, local block, Brauer character, and row in the Statement.

[F1]

Brauer's Second Main Theorem gives the required block-support implication for each row (Brauer's Second Main Theorem).

[F2]

Every ordinary irreducible belongs to one and only one global block (Blocks partition the ordinary and Brauer irreducible characters).

[F3]

The algebraically closed residue-field condition and AC are the hypotheses of F1 (An algebraically closed field: every nonconstant polynomial has a root in the field and The Axiom of Choice).

Proof

1.1

Apply F1 to the row $χ \in {Irr}_{K} (G, B)$ and the fixed local character $φ \in IBr (H, c)$ . A nonzero entry gives $c^{G} = B$ , proving the displayed implication.

F1F3

2.1

By F2 each row has a unique global block. Therefore any two nonzero rows in this column both belong to $c^{G}$ and cannot lie in distinct blocks. The argument permits the whole column to be zero and does not assert that any allowed entry is nonzero. The row set is finite; AC and algebraic closedness are used only through F1.

F1F2F3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Craven, The Brauer Correspondence, Theorem 2.22, pp. 29–30 (standard reference, not scraped)
Meierfrankenfeld, MTH 912 Class Notes, Theorem 6.7.15, pp. 169–171 (standard reference, not scraped)