Alphabeta Math
CorollaryStatement: AI-adaptedProof: Literature-sourcedPipeline-generatedaudited 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 local generalized-character criterion

Statement

Let F⊆C be a characteristic-zero splitting field for G. A class function f:G→F, viewed as complex-valued, is a generalized character if and only if f∣H is a generalized character for every elementary subgroup H of G.

Facts & Assumptions

[F1]

The cited prerequisite is Brauer induction.

Proof

Given: f∣H∈R(H) for every elementary H.

1.1

The forward implication is restriction stability. For the converse, choose the Brauer relation 1G=∑iniInd⁡HiGλi.

F1given
2.1

Pointwise multiplication and the projection formula give f=∑iniInd⁡HiG(λif∣Hi). Each inner product is in R(Hi), so the right side is in R(G). ∎

step 1.1

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources