Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-28
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.

FALSE: over every field, the endomorphism ring of an irreducible representation is just the base field

Statement

False claim. Over every field k, every irreducible representation of a group has endomorphism ring equal to k.

Facts & Assumptions

Given: The real 2-dimensional representation of C3 from the companion example.

[L1]

There is an irreducible real representation of C3 whose endomorphism ring is a copy of C (The real 2-dimensional irreducible representation of C3 has endomorphism ring C).

Refutation

technique · direct
1.1L1given

By [L1], over the field R there exists an irreducible representation whose endomorphism ring is C.

2.1step 1.1∎

Since C≠R, that representation contradicts the stated claim. Therefore the claim is false.

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