Donovan's conjecture on Morita finiteness of blocks with a given defect group
Let be a prime and an algebraically closed field of characteristic (or, in the integral form, a complete discrete valuation ring with residue field ). A block of has a defect group, a -subgroup determined up to conjugacy. Donovan's conjecture states that for a fixed finite -group , as ranges over all finite groups, the blocks with defect group isomorphic to fall into only finitely many Morita equivalence classes. Known cases include symmetric-group blocks (Scopes), abelian 2-groups (Eaton-Livesey), extraspecial groups of order for (An-Eaton) and quaternion defect groups (Eaton-Eisele-Kessar-Linckelmann-Schaeffer Fry, 2026). Does Donovan finiteness hold for every finite -group, every prime, over and over ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Modular representation theory of finite groups; block theory
- Posed by
- P. Donovan; recorded as Conjecture M in J. L. Alperin, Local representation theory, Santa Cruz Conference on Finite Groups, Proc. Sympos. Pure Math. 37 (1980)
- Year posed
- 1980
- Years open
- 46y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Field paper, Theorem 1.1: for each prime , algebraically closed of characteristic and bound , blocks of finite groups over with defect groups of order at most represent only finitely many -linear Morita classes; defect groups may be nonabelian and is included. Integral companion, Theorem 1.1 and Corollary: the same over and over each fixed complete DVR of characteristic zero with algebraically closed residue field, ramified included. Consequences: bounded Cartan entries and Loewy lengths. Not shown: explicit lists or bounds on the number of classes, or Morita finiteness over coefficient fields that are not algebraically closed.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems, published in the openai/math release (pinned commit adc7f12). The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it, and the README cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: abstracts, introductions and main theorems of both manuscripts (TeX source), read against Donovan's conjecture as recorded by Alperin. The field paper's Theorem 1.1 is the bounded-defect-order form over every algebraically closed field of characteristic (equivalent to the fixed-defect-group form, as the paper notes); the companion's Theorem 1.1 gives the integral form over and a corollary over every fixed complete characteristic-zero DVR with algebraically closed residue field. The field paper says its proof is independent of the integral companion, but its final coefficient-extension step recalls an argument from another unreviewed release manuscript (The Blockwise Alperin Weight Conjecture, Section 2.1). The proofs were not refereed. No Lean formalization.