Alphabeta Math
CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

The Schur index divides the representation degree

Statement

Let χ be an irreducible complex character, put K=Q(χ), and let V be the irreducible K-representation used in The Schur index of an irreducible character. Then mK(χ)∣dim⁡KV.

Facts & Assumptions

Given: χ, K, and V as in the statement, together with a finite Galois splitting field E/K inside C.

[L1]

dim⁡KV=mK(χ) [K(χ):K],χ(1) (Character formula over a nonsplitting field).

Proof

technique · direct
1.1

Since K(χ)=K, [L1] gives dim⁡KV=mK(χ)χ(1). The integer χ(1) is positive, so this expresses dim⁡KV as an integer multiple of mK(χ).

L1algebra
2.1

Hence mK(χ)∣dim⁡KV.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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