pith. sign in

arxiv: 2312.02594 · v5 · submitted 2023-12-05 · 🧮 math.RT

A reduction theorem for the Navarro Alperin weight conjecture

Zhicheng Feng , Qulei Fu , Yuanyang Zhou This is my paper

Pith reviewed 2026-05-24 05:29 UTC · model grok-4.3

classification 🧮 math.RT
keywords Navarro Alperin weight conjecturereduction theoremsimple groupsabelian Sylow 2-subgroupsGalois automorphismsfinite group representations
0
0 comments X

The pith

The Navarro Alperin weight conjecture reduces to simple groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper adapts an existing reduction theorem for the Alperin weight conjecture to its Navarro version, which requires compatibility with Galois automorphisms and group automorphisms. This shows that verifying the conjecture on finite simple groups is sufficient for all finite groups. As an application, the authors establish the conjecture for all finite groups that have abelian Sylow 2-subgroups. A sympathetic reader would care because this narrows the problem to checking simple groups and resolves it in an important special case.

Core claim

We give a reduction to simple groups for the Navarro Alperin weight conjecture. As an application, we prove the conjecture for the finite groups with abelian Sylow 2-subgroups.

What carries the argument

The reduction theorem that extends the Navarro-Tiep reduction to account for Galois and outer automorphisms.

Load-bearing premise

The Navarro-Tiep reduction for the ordinary Alperin weight conjecture adapts to the Navarro version without new obstructions from the added Galois and automorphism conditions.

What would settle it

A finite simple group in which the Navarro Alperin weight conjecture fails would show either that the adaptation does not hold or that the reduction theorem is incorrect.

read the original abstract

The Alperin weight conjecture has been reduced to simple groups by Navarro and Tiep. In this paper, we investigate the Navarro Alperin weight conjecture, which includes Galois automorphisms and group automorphisms in comparison with the original version, and give a reduction to simple groups. As an application, we prove the conjecture for the finite groups with abelian Sylow 2-subgroups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes a reduction of the Navarro Alperin weight conjecture (incorporating Galois automorphisms and group automorphisms) to the case of finite simple groups. The argument adapts the Navarro-Tiep reduction for the ordinary Alperin weight conjecture. As an application, the conjecture is verified for all finite groups whose Sylow 2-subgroups are abelian.

Significance. If the reduction holds, the result is a useful extension of existing reduction theorems in the area, narrowing the Navarro version of the conjecture to simple groups and supplying an explicit infinite family of groups for which the conjecture is now known to hold.

minor comments (2)
  1. [Introduction] The introduction would benefit from a brief explicit comparison (e.g., a short table or enumerated list) of the precise additional data (Galois action, outer automorphisms) that must be preserved in the reduction relative to the Navarro-Tiep argument.
  2. Notation for the action of Galois automorphisms on weights and on the set of irreducible characters should be introduced once and used consistently; several passages reuse the same symbol for distinct maps.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance as an extension of the Navarro-Tiep reduction, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; reduction adapts external prior result

full rationale

The paper states a reduction to simple groups for the Navarro Alperin weight conjecture by adapting the Navarro-Tiep reduction (different authors) for the ordinary Alperin weight conjecture. The application to groups with abelian Sylow 2-subgroups follows directly. No self-definitional steps, no fitted inputs renamed as predictions, and no load-bearing self-citation chains appear in the abstract or described argument. The derivation remains self-contained against the external benchmark of the prior reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure-mathematics reduction theorem operating inside the standard axiomatic framework of finite groups and modular representation theory; no new entities or fitted parameters are introduced.

axioms (1)
  • standard math Standard axioms and theorems of finite group theory and p-modular representation theory
    The reduction and application are carried out within the established body of results on finite groups, blocks, and characters.

pith-pipeline@v0.9.0 · 5582 in / 1237 out tokens · 35010 ms · 2026-05-24T05:29:24.183835+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A reduction theorem for the blockwise Navarro Alperin weight conjecture via H-triples

    math.GR 2025-12 unverdicted novelty 6.0

    A reduction theorem establishes that the blockwise Galois Alperin weight conjecture follows from inductive conditions on simple groups via isomorphisms of H-triples.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · cited by 1 Pith paper

  1. [1]

    J. L. Alperin, Local Representation Theory . Cambridge Stud. Adv. Math., vol. 11, Cam- bridge University Press, Cambridge, 1986

    work page 1986

  2. [2]

    J. L. Alperin, Weights for finite groups. In: The Arcata Conference on Representations of Finite Groups, Arcata, Calif. (1986), Part I . Proc. Sympos. Pure Math., vol. 47, pp. 369–379, Am. Math. Soc., Providence, 1987

    work page 1986

  3. [3]

    J. An, H. Dietrich, The A WC-goodness and essential rank o f sporadic simple groups. J. Algebra 356 (2012), 325–354

    work page 2012

  4. [4]

    Borel, J

    A. Borel, J. Tits, ´El´ ements unipotents et sous-groupes paraboliques de groupes r´ eductifs. I. Invent. Math. 12 (1971), 95–104

    work page 1971

  5. [5]

    Brough, B

    J. Brough, B. Sp¨ ath, A criterion for the inductive Alper in weight condition. Bull. Lond. Math. Soc. 54 (2022), 466–481

    work page 2022

  6. [6]

    Burkhardt, Die Zerlegungsmatrizen der Gruppen PSL(2 ,p f )

    R. Burkhardt, Die Zerlegungsmatrizen der Gruppen PSL(2 ,p f ). J. Algebra 40 (1976), 75– 96. 38

    work page 1976

  7. [7]

    Cabanes, M

    M. Cabanes, M. Enguehard, Representation Theory of Finite Reductive Groups . New Math. Monogr., vol. 1, Cambridge University Press, Cambridge, 2004

    work page 2004

  8. [8]

    Cabanes, B

    M. Cabanes, B. Sp¨ ath, Equivariant character correspon dences and inductive McKay con- dition for type A. J. reine angew. Math. 728 (2017), 153–194

    work page 2017

  9. [9]

    Cabanes, B

    M. Cabanes, B. Sp¨ ath, The McKay Conjecture on character degrees. preprint

    work page

  10. [10]

    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A . Wilson, Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups . Oxford University Press, Eynsham, 1985

    work page 1985

  11. [11]

    C. W. Curtis, Modular representations of finite groups w ith split (B,N )-pairs. In: Seminar on Algebraic Groups and Related Finite Groups (The Institute f or Advanced Study, Prince- ton, N.J., 1968/69) , pp. 57–95, Lecture Notes in Math., vol. 131, Springer, Berlin–New York, 1970

    work page 1968

  12. [12]

    E. C. Dade, Counting characters in blocks with cyclic de fect groups. I. J. Algebra 186 (1996), 934–969

    work page 1996

  13. [13]

    Denoncin, Stable basic sets for finite special linear and unitary groups

    D. Denoncin, Stable basic sets for finite special linear and unitary groups. Adv. Math. 307 (2017), 344–368

    work page 2017

  14. [14]

    Feit, Extending Steinberg characters

    W. Feit, Extending Steinberg characters. In: Linear Algebraic Groups and Their Represen- tations (Los Angeles, CA, 1992) , pp. 1–9, Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993

    work page 1992

  15. [15]

    Z. Feng, C. Li, J. Zhang, Equivariant correspondences a nd the inductive Alperin weight condition for type A. Trans. Amer. Math. Soc. 374 (2021), 8365–8433

    work page 2021

  16. [16]

    Z. Feng, C. Li, J. Zhang, On the inductive blockwise Alpe rin weight condition for type A. J. Algebra 631, (2023) 287–305

    work page 2023

  17. [17]

    Z. Feng, Z. Li, J. Zhang, On the inductive blockwise Alpe rin weight condition for classical groups. J. Algebra 537 (2019), 381–434

    work page 2019

  18. [18]

    Z. Feng, B. Sp¨ ath, Unitriangular basic sets, Brauer ch aracters and coprime actions. Repre- sent. Theory 27 (2023), 115–148

    work page 2023

  19. [19]

    Z. Feng, J. Zhang, Alperin weight conjecture and relate d developments. Bull. Math. Sci. 12 (2022), 2230005

    work page 2022

  20. [20]

    Fu, An equivariant bijection of irreducible Brauer c haracters above the Dade– Glauberman–Nagao correspondence

    Q. Fu, An equivariant bijection of irreducible Brauer c haracters above the Dade– Glauberman–Nagao correspondence. arXiv:2311.06471

    work page arXiv

  21. [21]

    M. Geck, G. Hiss, F. L¨ ubeck, G. Malle, G. Pfeiffer, CHEVIE – a system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras, Appl. Algebra Eng. Commun. Comput. 7 (1996), 175–210. 39

    work page 1996

  22. [22]

    Gorenstein, R

    D. Gorenstein, R. Lyons, R. Solomon, The Classification of the Finite Simple Groups, Number 3 . Math. Surveys Monogr., vol. 40, American Mathematical Society, Providence, RI, 1998

    work page 1998

  23. [23]

    I. M. Isaacs, Characters of π -separable groups. J. Algebra 86 (1984), 98–128

    work page 1984

  24. [24]

    I. M. Isaacs, G. Malle, G. Navarro, A reduction theorem f or the McKay conjecture. Invent. Math. 170 (2007), 33–101

    work page 2007

  25. [25]

    Jansen, K

    C. Jansen, K. Lux, R. Parker, R. Wilson, An Atlas of Brauer Characters . London Mathe- matical Society Monographs. New Series, vol. 11. Oxford Science Publications. The Claren- don Press, Oxford University Press, New York, 1995

    work page 1995

  26. [26]

    Johansson, On the inductive McKay–Navarro conditio n for finite groups of Lie type in their defining characteristic

    B. Johansson, On the inductive McKay–Navarro conditio n for finite groups of Lie type in their defining characteristic. J. Algebra 610 (2022), 223–240

    work page 2022

  27. [27]

    Johansson, The inductive McKay–Navarro condition for finite groups of Lie type

    B. Johansson, The inductive McKay–Navarro condition for finite groups of Lie type. Dis- sertation, TU Kaiserslautern, 2022

    work page 2022

  28. [28]

    Johansson, The inductive McKay–Navarro condition f or the Suzuki and Ree groups and for groups with non-generic Sylow normalizers

    B. Johansson, The inductive McKay–Navarro condition f or the Suzuki and Ree groups and for groups with non-generic Sylow normalizers. J. Pure Appl. Algebra 228 (2024), 107469

    work page 2024

  29. [29]

    P. B. Kleidman, The subgroup structure of some finite simple groups. Ph.D. Thesis, Trinity College, 1986

    work page 1986

  30. [30]

    Landrock, G

    P. Landrock, G. O. Michler, Principal 2-blocks of the si mple groups of Ree type. Trans. Amer. Math. Soc. 260 (1980), 83–111

    work page 1980

  31. [31]

    Malle, B

    G. Malle, B. Sp¨ ath, Characters of odd degree. Ann. of Math. (2) 184 (2016), 869–908

    work page 2016

  32. [32]

    J. M. Mart ´ ınez, N. Rizo, D. Rossi, The Alperin weight co njecture and the Glauberman correspondence via character triples. arXiv:2311.05536

    work page arXiv

  33. [33]

    Nagao, Y

    H. Nagao, Y. Tsushima, Representations of Finite Groups . Academic Press Inc., Boston, 1989

    work page 1989

  34. [34]

    Navarro, Weights, vertices, and a correspondence of character in groups of odd order

    G. Navarro, Weights, vertices, and a correspondence of character in groups of odd order. Math. Z. 212 (1993), 535–544

    work page 1993

  35. [35]

    Navarro, The Mckay conjecture and Galois automorphi sms

    G. Navarro, The Mckay conjecture and Galois automorphi sms. Ann. of Math. (2) 160 (2004), 1129–1140

    work page 2004

  36. [36]

    Navarro, Character Theory and the McKay Conjecture

    G. Navarro, Character Theory and the McKay Conjecture . Cambridge Stud. Adv. Math., vol. 175, Cambridge University Press, Cambridge, 2018

    work page 2018

  37. [37]

    Navarro, Decomposition matrices over unitriangula rs

    G. Navarro, Decomposition matrices over unitriangula rs. J. Algebra 619 (2023), 572–582

    work page 2023

  38. [38]

    Navarro, B

    G. Navarro, B. Sp¨ ath, C. Vallejo, A reduction theorem f or the Galois–Mckay conjecture. Trans. Amer. Math. Soc. 373 (2020), 6157–6183

    work page 2020

  39. [39]

    Navarro, P

    G. Navarro, P. H. Tiep, A reduction theorem for the Alper in weight conjecture. Invent. Math. 184 (2011), 529–565. 40

    work page 2011

  40. [40]

    Ruhstorfer, A

    L. Ruhstorfer, A. A. Schaeffer Fry, The inductive McKay–N avarro conditions for the prime 2 and some groups of Lie type. Proc. Amer. Math. Soc. Ser. B 9 (2022), 204–220

    work page 2022

  41. [41]

    Sp¨ ath, Inductive conditions for counting conjectu res via character triples

    B. Sp¨ ath, Inductive conditions for counting conjectu res via character triples. In: Represen- tation Theory – Current Trends and Perspectives . EMS Ser. Congr. Rep., Eur. Math. Soc., Z¨ urich, 2017, pp. 665–680

    work page 2017

  42. [42]

    Srinivasan, C

    B. Srinivasan, C. R. Vinroot, Galois group action and Jo rdan decomposition of characters of finite reductive groups with connected center. J. Algebra 558 (2020), 708–727

    work page 2020

  43. [43]

    Turull, The strengthened Alperin weight conjecture for p-solvable groups

    A. Turull, The strengthened Alperin weight conjecture for p-solvable groups. J. Algebra 398 (2014), 469–480

    work page 2014

  44. [44]

    J. H. Walter, The characterization of finite groups with abelian Sylow 2-subgroups. Ann. of Math. (2) 89 (1969), 405–514

    work page 1969

  45. [45]

    H. N. Ward, On Ree’s series of simple groups. Trans. Amer. Math. Soc. 121 (1966), 62–89

    work page 1966

  46. [46]

    Webb, A Course in Finite Group Representation Theory

    P. Webb, A Course in Finite Group Representation Theory . Cambridge Stud. Adv. Math., vol. 161, Cambridge University Press, Cambridge, 2016. 41

    work page 2016