Over F_5, the source's rank-one algebra H_q=F_5[T]/((T-q)(T+1)) is local at q=-1 and split semisimple at q=1. A projective global-to-generic construction from stable basic sets, integral pinned Harish-Chandra data, actual graded Clifford algebras, and a strictly Brauer-compatible BDR transport remains viable within the source's stated hypotheses.
Route status · Narrowed routeFinite groups · modular representation theory · block theory · local representation theory
Alperin Weight Conjecture
Collaboration betaThe source reports substantial reusable reductions and projective-algebra tools, but no nontrivial quasi-isolated block has yet been completed with all stabilizer, scalar, and intermediate-block requirements, and full prime-and-cover coverage remains open.
Known results and sources
Research problem
Exact mathematical statement
For every finite group G, prime p, and p-block B, the number l(B) of irreducible Brauer characters in B equals the number w(B) of G-orbits of B-weights:
The classification route in the source targets a stronger inductive condition: an equivariant bijection between Brauer characters and weights together with the exact block isomorphism of modular character triples. The governing Markdown explicitly does not claim that the covering-group hypotheses needed for the final reduction have been verified.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Alperin Weight Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
The source reports a checked literature reduction from the exact inductive modular-character-triple condition for all required covering groups to arbitrary finite groups; a numerical count for simple quotients is not enough.
Evidence posture · Source-reported route statement · dependencies incompleteWork mapped so far
Alperin Weight Conjecture in numbers
1kretained lines of mathematical investigation1,000 in the current working snapshot
- Argument development
- 745 · 75%
- Explored or eliminated routes
- 32 · 3%
- Computational analysis
- 38 · 4%
- Open obligations
- 43 · 4%
- Definitions and setup
- 142 · 14%
8selected mapped statements1routes investigated4open questions4contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Prove the exact Corollary 6.10 global-to-generic block-triple relation for every label of one nontrivial quasi-isolated block.
Suggested move: Choose one quasi-isolated block satisfying Condition 6.1 and build a row for every global label recording the generic weight, inertia groups, oriented Clifford algebra, projective label, central scalars, and every intermediate block induction.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
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Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Explored alternatives
Other routes
Over F_5, the source's rank-one algebra H_q=F_5[T]/((T-q)(T+1)) is local at q=-1 and split semisimple at q=1. A projective global-to-generic construction from stable basic sets, integral pinned Harish-Chandra data, actual graded Clifford algebras, and a strictly Brauer-compatible BDR transport remains viable within the source's stated hypotheses.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Prove the exact Corollary 6.10 global-to-generic block-triple relation for every label of one nontrivial quasi-isolated block.Suggested move: Choose one quasi-isolated block satisfying Condition 6.1 and build a row for every global label recording the generic weight, inertia groups, oriented Clifford algebra, projective label, central scalars, and every intermediate block induction.
Ready to work on
Verify a concrete outer-graded BDR equivalence and the exact equivalence-induced versus Brauer-induced Dade maps, including the intermediate-block effect of any H^1 gauge.Suggested move: For one exact BDR reduction, write the actual global and local homogeneous maps on the Dade crossed products and test strict equality; if a gauge appears, prove its effect on every common-centralizer scalar and Harris-Knorr block.
Ready to work on
One complete quasi-isolated block
Select one nontrivial quasi-isolated block and prove every clause of the exact Corollary 6.10 input label by label, including all intermediate block inductions.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Replace broad family summaries by exact iBAW certificates for every required covering group and every excluded-prime or exceptional-cover sector.Suggested move: Upgrade one claimed family in the coverage matrix to an exact theorem certificate, then enumerate and close every prime, cover, automorphism, and exceptional sector outside Condition 6.1.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
The full Alperin weight conjecture remains open. A 2026 peer-reviewed article reports inductive-condition verification for alternating groups, sporadic groups, groups of Lie type in defining characteristic, and type A, while stating that types D and E remain largely open. These family-level advances do not establish the conjecture for every finite group and block.
[5]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Peer reviewedFeng, Malle, and Zhang introduced generic weights for finite reductive groups and derived criteria for inductive Alperin weight conditions; the article presents this as progress toward, not a proof of, the…[5] Peer reviewedAn, Hiss, and Lübeck verified the inductive blockwise Alperin weight condition in odd nondefining characteristic for the finite groups F4(q).[4] Peer reviewedSpäth established a reduction theorem for the blockwise version, conditional on inductive conditions for every finite nonabelian simple group.[3] Peer reviewedNavarro and Tiep reduced the non-blockwise Alperin weight conjecture to an inductive condition for finite simple groups and proved several cases.[2]
Mathematical neighborhood
Related results and reusable starting points
Current focusAlperin Weight Conjecture
The non-blockwise conjecture for arbitrary finite groups follows from the stated inductive Alperin weight condition for finite nonabelian simple groups.
[2]The blockwise conjecture is reduced to a corresponding set of inductive conditions for every finite nonabelian simple group.
[3]The inductive blockwise condition is verified for F4(q) in odd characteristic not dividing q; this is a family-level case, not the universal conjecture.
[4]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA statement-aligned formalization must define finite-group p-blocks, irreducible Brauer characters, radical p-subgroups, block induction, defect-zero characters, and block weights under one fixed convention.
- Formalization targetA formal reduction must distinguish the numerical blockwise equality from the stronger equivariant inductive conditions and must preserve automorphism and block compatibility.
- Formalization targetFamily-level certificates must not be promoted to a proof for all finite nonabelian simple groups or all covering groups and primes.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
6 standing statements2 proposed statements4 open questions1 narrowed routes
- theorem candidate
1 of 8 1 - reduction
2 of 8 2 - lemma
3 of 8 3 - special case
1 of 8 1 - negative result
1 of 8 1
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
- retained route statementDoes every block contain as many irreducible Brauer characters as it has conjugacy classes of weights?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementCovering-group reductionintermediate
- retained route statementNormal-p quotient packet identitiesintermediate
- retained route statementNormal-defect block correspondenceintermediate
- retained route statementExtension-free projective Clifford labelsintermediate
- retained route statementAbstract gauge does not settle block transportintermediate
- Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
- Useful failureTransport modular blocks by deforming a Hecke parameter to q=1reported failure
- Research targetProve the exact Corollary 6.10 global-to-generic block-triple relation for every label of one nontrivial quasi-isolated block.open
- Research targetVerify a concrete outer-graded BDR equivalence and the exact equivalence-induced versus Brauer-induced Dade maps, including the intermediate-block effect of any H^1 gauge.open
- Research targetReplace broad family summaries by exact iBAW certificates for every required covering group and every excluded-prime or exceptional-cover sector.open
- Research targetOne complete quasi-isolated blockopen
- Narrowed routeTransport modular blocks by deforming a Hecke parameter to q=1Over F_5, the source's rank-one algebra H_q=F_5[T]/((T-q)(T+1)) is local at q=-1 and split semisimple at q=1. A projective global-to-generic construction from stable basic sets, integral pinned Harish-Chandra data, actual graded Clifford algebras, and a strictly Brauer-compatible BDR transport remains viable within the source's stated hypotheses.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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Prepared starting pointProve the exact Corollary 6.10 global-to-generic block-triple relation for every label of one nontrivial quasi-isolated block.
Alperin Weight Conjecture · ready to start
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Research contextPrepared context for any AI agent
The source reports substantial reusable reductions and projective-algebra tools, but no nontrivial quasi-isolated block has yet been completed with all stabilizer, scalar, and intermediate-block requirements, and full prime-and-cover coverage remains open.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references5 cited works · next context review by Nov 30, 2026
The mathematical context was checked on Aug 30, 2026. Status can be refreshed sooner after a material result or claim.
- 1Weights for finite groupsoriginal source · J. L. Alperin · American Mathematical Society · 1987 · DOI 10.1090/pspum/047.1/933373 · MR MR933373 · accessed Aug 30, 2026
- 3A reduction theorem for the blockwise Alperin weight conjecturepeer reviewed result · Britta Späth · Journal of Group Theory · 2013-03-01 · DOI 10.1515/jgt-2012-0032 · accessed Aug 30, 2026
- 4The Inductive Blockwise Alperin Weight Condition for the Chevalley Groups F4(q)peer reviewed result · Jianbei An, Gerhard Hiss, Frank Lübeck · Memoirs of the American Mathematical Society · 2024-12-10 · DOI 10.1090/memo/1530 · accessed Aug 30, 2026
- 5Generic weights for finite reductive groupspeer reviewed result · Zhicheng Feng, Gunter Malle, Jiping Zhang · Mathematische Annalen · 2026-03-05 · ARXIV 2505.22064 · DOI 10.1007/s00208-026-03415-7 · accessed Aug 30, 2026
Important qualifications
- This was a bounded statement-aligned search of original, peer-reviewed, and publisher sources, not an exhaustive literature, priority, rights, authorship, or citation review.
- Open status is bound to a peer-reviewed article published on 2026-03-05 that explicitly says the conjecture remains open; this does not rule out a later claim after that date.
- The cited family-level verifications do not establish the conjecture for all finite groups or all blocks.
- No packet attachment was opened, and no URL was discovered or selected from packet contents; every public source was located through an independent search.
- The bounded search did not establish a statement-aligned formalization, certificate, dataset, or independently reproduced computation. Empty readiness lists do not prove that none exists.
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