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The Blockwise Alperin Weight Conjecture
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Abstract
We prove the numerical blockwise Alperin weight conjecture for every finite group and every prime p. For each p-block B, the number of irreducible Brauer characters in B equals the number of conjugacy classes of B-weights.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- Baoyu Zhang, ModularRep and the manuscript application (GitHub, accessed 2026-10-21). This Lean companion to Zhang's work cited in the manuscript treats the inductive blockwise condition for types B and C and sporadic groups under explicit external assumptions whose joint satisfiability the repository says has not been established; it does not formalize OpenAI's general numerical theorem.
Possibly relevant literature (found by the agent, not checked)
- Geoffrey R. Robinson, Elements of the Green ring related to Alperin's weight conjecture (arXiv:2610.27988, 2026-10-20). Restates the numerical questions of the weight conjecture in module form in the Green ring of RG, the same numerical equality that Theorem 1.1 of the paper asserts; it does not mention the release.
- J. Miquel Martínez, Noelia Rizo and Damiano Rossi, The Alperin Weight Conjecture and the Glauberman correspondence via character triples (arXiv:2311.05536, 2023; Algebra & Number Theory 20 (2026) 333–382). Studies Navarro's refinement and reduces inductive conditions to an automorphism-equivariant weight correspondence, in contrast to the paper's stated restriction to a numerical equality without a canonical or equivariant correspondence.
- Zhicheng Feng, Qulei Fu, Yuanyang Zhou, A reduction theorem for the blockwise Navarro Alperin weight conjecture via H-triples (arXiv:2512.15243, 2025-12-17, revised 2026-04-24). Formulates and reduces a blockwise Galois version of the weight conjecture, a refinement of the blockwise numerical statement l(B)=|W_p(B)| proved in the paper.
- Xin Huang, The Galois Alperin weight conjecture for finite category algebras (arXiv:2604.06166, 2026-04-07, revised 2026-05-07). Extends Galois and blockwise weight conjectures to finite category algebras and reduces them to finite groups, the setting of the paper's main theorem.
Digests and expositions
- AI Proved What?, Global = local? The blockwise Alperin weight conjecture (episode 202, October 2026; status checked 2026-10-19). This video and transcript explain the algebraic reduction and curve-cover strategy and report illustrative computations in known small cases, while explicitly stating that the geometric argument was not checked. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 202
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Blockwise-Alperin-Weight-Conjecture-September-23-2026,
author = {{OpenAI}},
title = {{The Blockwise Alperin Weight Conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf}{OAI:The-Blockwise-Alperin-Weight-Conjecture-September-23-2026}},
year = {2026}
}Family 202: The blockwise Alperin weight conjecture
Proves the numerical blockwise Alperin weight conjecture for every prime and every finite group: the number of irreducible Brauer characters in a block equals the number of conjugacy classes of its weights.
This is the only paper in this family.