The blockwise Alperin weight conjecture
Let be a prime and a finite group. A -weight of is a pair with a -subgroup and an irreducible character of of -defect zero; each weight is assigned to a -block of by Brauer induction. For a block , let be the number of irreducible Brauer characters (simple modules) in and let be the set of -conjugacy classes of weights assigned to . Alperin conjectured that simple modules are counted locally, block by block. Does hold for every prime , every finite group and every -block ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Modular representation theory of finite groups
- Posed by
- J. L. Alperin, Weights for finite groups (Arcata conference 1986, proceedings 1987)
- Year posed
- 1987
- Years open
- 39y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 55 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims for every prime , finite group and -block , including the case. Consequences derived: for the Brauer correspondent , equality for abelian defect groups, the Sylow normalizer inequality for Brauer character counts, and principal-block bounds of Hung, Sambale and Tiep. The proof uses chain inversion and powers modulo commutators to reduce to counting tuples in groups with trivial , then proves the needed -power divisibility via moduli of marked genus-one covers. It is an equality of cardinalities only: no natural or equivariant bijection, and no inductive condition for simple groups, is established.
What the AI did
The release README says the manuscripts were produced by an unreleased internal OpenAI model, which was posed some 4,000 open research problems during an evaluation, with the vast majority of results obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each. This manuscript is not among the README's named exceptions. It is credited to OpenAI with no human author named.
Verification
No independent mathematician has checked this yet. Theorem 1 was read against the posed conjecture and states the numerical blockwise form exactly, with no restriction on prime, group or defect group, using Brauer's central-character block induction. There is no Lean formalization for this manuscript in the release. Nothing was re-run here. A reader should know the proof route is unusual: it avoids the classification of finite simple groups and the inductive blockwise weight condition, reducing instead to a divisibility statement for tuple counts proved through high-index genus-one curve covers and inseparable genus change in characteristic p. The equivariant or canonical refinements of the conjecture are not claimed.