Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: 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.

Brauer induction for S3

Example

Let T≅C2 and A≅C3 in S3, let ϵ be the sign character, and let ρ be the degree-two irreducible. If ω is either nontrivial linear character of A, then ρ=Ind⁡AS3ω, 1S3=Ind⁡TS31T−Ind⁡AS3ω,ϵ=Ind⁡TS3sgn⁡T−Ind⁡AS3ω.

Facts & Assumptions

[F1]

The cited prerequisite is Frobenius' formula for the character of an induced representation.

Verification

Given: the values of 1,ϵ,ρ on the classes 1,(12),(123).

1.1F1given

Frobenius' formula gives Ind⁡T1T=1+ρ, Ind⁡Tsgn⁡T=ϵ+ρ, and Ind⁡Aω=ρ by evaluating on those three classes.

2.1step 1.1

Subtract the final equality from the first two. Both T and A are elementary, so these are integral Brauer-induction expressions. ∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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