CorollaryStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Elementary restriction detects generalized characters

Statement

The map $R (G) \to \prod_{H \in E} R (H)$ given by restriction to all elementary subgroups is injective.

Facts & Assumptions

[F1]

The cited prerequisite is Brauer induction.

Proof

Given: $χ \in R (G)$ restricts to $0$ on every elementary subgroup.

1.1

Brauer induction writes $1_{G} = \sum_{i} n_{i} {Ind}_{H_{i}}^{G} λ_{i}$ with $H_{i}$ elementary and $λ_{i}$ linear.

F1given

2.1

The projection formula yields $χ = \sum_{i} n_{i} {Ind}_{H_{i}}^{G} (λ_{i} \cdot {Res}_{H_{i}}^{G} χ) = 0$ , as every restriction is zero. ∎

step 1.1

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Tammo tom Dieck, Representation Theory, Proposition 4.6.8 (standard reference, not scraped)