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arxiv: 2605.22969 · v1 · pith:TXBHA2YSnew · submitted 2026-05-21 · 🧮 math.RT · math.GR

Real 2-blocks in quasi-simple groups

John Revere McHugh , A. A. Schaeffer Fry This is my paper

Pith reviewed 2026-05-25 05:20 UTC · model grok-4.3

classification 🧮 math.RT math.GR
keywords quasi-simple groups2-blocksBrauer charactersquadratic typeMathieu group M22complex conjugationreal representations
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The pith

The Mathieu group M22 is the only simple group without a nontrivial irreducible Brauer character of quadratic type.

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

The paper classifies all quasi-simple groups that admit a non-principal 2-block stable under complex conjugation. From this classification it follows directly that every simple group except M22 possesses at least one nontrivial irreducible Brauer character of quadratic type. The result answers a question raised by Gow and Murray on the existence of such characters in the 2-modular representation theory of finite simple groups. A reader cares because the presence or absence of quadratic-type characters controls the reality properties of modular representations and the structure of real blocks.

Core claim

The authors determine precisely which quasi-simple groups possess a non-principal 2-block that remains invariant under complex conjugation. As a corollary they establish that M22 is the unique simple group lacking any nontrivial irreducible Brauer character of quadratic type.

What carries the argument

A non-principal 2-block stable under complex conjugation; its invariance forces the existence of real Brauer characters of quadratic type in the corresponding block.

If this is right

  • Every simple group other than M22 has at least one nontrivial real Brauer character of quadratic type.
  • The only quasi-simple exceptions to the block-stability property are known and finite in number.
  • The reality of 2-blocks is completely determined once the classification and prior block results are granted.
  • The quadratic-type condition distinguishes M22 from all other simple groups in characteristic 2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same classification technique could be applied to odd primes to decide which groups admit real p-blocks of quadratic type.
  • The result constrains possible constructions of real representations or self-dual modules over fields of characteristic 2.
  • If a new quasi-simple group were found outside the known list, its 2-blocks would have to be checked separately against the stability criterion.

Load-bearing premise

Every quasi-simple group and its 2-block structure is already accounted for by the classification of finite simple groups together with known results on blocks of groups of Lie type and sporadic groups.

What would settle it

An explicit computation or new example showing that M22 possesses a nontrivial irreducible Brauer character of quadratic type, or the discovery of another simple group whose 2-blocks contradict the listed exceptions.

read the original abstract

We determine which quasi-simple groups have a non-principal $2$-block that is stable under complex conjugation. As a corollary, we determine that the Mathieu group $M_{22}$ is the only simple group not possessing a nontrivial irreducible Brauer character of quadratic type, answering a recent question of Gow and Murray.

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

2 major / 2 minor

Summary. The manuscript classifies all quasi-simple groups that possess a non-principal 2-block stable under complex conjugation. As a corollary it concludes that the Mathieu group M_{22} is the only simple group lacking a nontrivial irreducible Brauer character of quadratic type, thereby answering a question of Gow and Murray.

Significance. If correct, the classification supplies a definitive answer to the existence question for quadratic-type Brauer characters across all simple groups. The argument rests on the classification of finite simple groups together with previously established results on 2-blocks of groups of Lie type and sporadic groups; the manuscript therefore inherits the strengths (and any limitations) of those external theorems once the conjugation-stability condition is verified to apply verbatim.

major comments (2)
  1. [§4] §4 (groups of Lie type in non-defining characteristic): the claim that every cited block theorem automatically yields a conjugation-stable block requires an explicit check that the Galois action of complex conjugation preserves the block idempotent; without this verification the reduction for twisted groups (e.g., ^2E_6(q) or ^3D_4(q)) remains incomplete.
  2. [§5] §5 (defining characteristic 2): several small-rank cases (e.g., SL_3(2), Sp_4(2)') are dispatched by direct citation; the manuscript must confirm that the external 2-block classifications used there already incorporate the real-block condition rather than merely listing blocks.
minor comments (2)
  1. [Introduction] Notation for the conjugation automorphism is introduced only in the introduction; repeating the definition at the start of each case-analysis section would improve readability.
  2. [Table 1] Table 1 (list of exceptions) omits the precise reference for each sporadic group; adding a column with the cited theorem would make the table self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the classification of conjugation-stable non-principal 2-blocks. We address each major point below with targeted revisions to strengthen the arguments without altering the main results.

read point-by-point responses

  1. Referee: [§4] §4 (groups of Lie type in non-defining characteristic): the claim that every cited block theorem automatically yields a conjugation-stable block requires an explicit check that the Galois action of complex conjugation preserves the block idempotent; without this verification the reduction for twisted groups (e.g., ^2E_6(q) or ^3D_4(q)) remains incomplete.

    Authors: We agree that an explicit verification strengthens the reduction. In the revised manuscript we will insert a short paragraph in §4 explaining that the Galois action of complex conjugation (corresponding to the nontrivial element of Gal(Q(ζ)/Q) where ζ is a suitable root of unity) preserves the block idempotents for the cited theorems on groups of Lie type. This follows from the fact that the block idempotents are rational linear combinations of class sums and that the relevant Brauer characters are fixed by the conjugation automorphism in the sources (e.g., the results of Cabanes–Enguehard and others already ensure the blocks are defined over Q or are stable under the action). We will explicitly note the cases ^2E_6(q) and ^3D_4(q) to confirm the twisted groups are covered verbatim. revision: yes

  2. Referee: [§5] §5 (defining characteristic 2): several small-rank cases (e.g., SL_3(2), Sp_4(2)') are dispatched by direct citation; the manuscript must confirm that the external 2-block classifications used there already incorporate the real-block condition rather than merely listing blocks.

    Authors: The external classifications (e.g., those of An–Hiss and others for small-rank groups in characteristic 2) list all 2-blocks together with their defect groups and Brauer characters, from which the conjugation-stable ones are immediately identifiable because the character tables and block idempotents are given explicitly over the rationals or are known to be real. Nevertheless, to meet the referee’s request we will add a clarifying sentence in §5 stating that the cited sources already encode the real-block condition via the explicit determination of the blocks and their associated irreducible Brauer characters, rather than providing only an unordered list. revision: yes

Circularity Check

0 steps flagged

No circularity; central claim rests on external CFSG reduction and prior independent block theorems

full rationale

The derivation reduces the problem to a finite list of quasi-simple groups via the classification of finite simple groups, then applies previously published theorems on 2-blocks of Lie-type and sporadic groups to identify those with conjugation-stable non-principal blocks. These supporting results are external and independent of the present paper; no equation or definition within the paper is shown to be equivalent to its own inputs by construction, and no load-bearing step reduces to a self-citation chain or fitted parameter renamed as a prediction. The corollary identifying M_{22} follows directly from this case analysis without internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the background assumptions typical for classification results in this area; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption The classification of finite simple groups and the known descriptions of 2-blocks for groups of Lie type and sporadic groups are complete and correctly applied.
    The reduction to checking known families of quasi-simple groups relies on this background classification.

pith-pipeline@v0.9.0 · 5565 in / 1300 out tokens · 30924 ms · 2026-05-25T05:20:55.396180+00:00 · methodology

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Brou\' e

    M. Brou\' e . Les -blocs des groupes GL(n,q) et U(n,q^2) et leurs structures locales. Ast\' e risque , 133--134:159--188, 1986

    work page 1986

  2. [2]

    Brou\' e and J

    M. Brou\' e and J. Michel. Blocs et s\' e ries de L usztig dans un groupe r\' e ductif fini. J. reine angew. Math. , 395:56--67, 1989

    work page 1989

  3. [3]

    Cabanes and M

    M. Cabanes and M. Enguehard. Representation theory of finite reductive groups , volume 1 of New Mathematical Monographs . Cambridge University Press, Cambridge, 2004

    work page 2004

  4. [4]

    R. Carter. Finite Groups of Lie Type: Conjugacy Classes and Complex Characters . Wiley-Interscience, New York, 1985

    work page 1985

  5. [5]

    Chang and R

    B. Chang and R. Ree. The characters of G 2 (q) . In Symposia M athematica, V ol. XIII ( C onvegno di G ruppi A beliani & C onvegno di G ruppi e loro R appresentazioni, INDAM , R ome, 1972) , pages 395--413. Academic Press, London-New York, 1974

    work page 1972

  6. [6]

    D. I. Deriziotis and G. O. Michler. Character table and blocks of finite simple triality groups ^3 D _4(q) . Trans. Amer. Math. Soc. , 303(1):39--70, 1987

    work page 1987

  7. [7]

    Digne and J

    F. Digne and J. Michel. On L usztig's parametrization of characters of finite groups of L ie type. Ast\' e risque , 181--182:113--156, 1990

    work page 1990

  8. [8]

    H. Enomoto. The characters of the finite C hevalley group G 2 (q),q=3 f . Japan. J. Math. (N.S.) , 2(2):191--248, 1976

    work page 1976

  9. [9]

    P. Fong. On decomposition numbers of J 1 and R(q) . In Symposia M athematica, V ol. XIII ( C onvegno di G ruppi A beliani & C onvegno di G ruppi e loro R appresentazioni, INDAM , R ome, 1972) , pages 415--422. Academic Press, London-New York, 1974

    work page 1972

  10. [10]

    Fong and B

    P. Fong and B. Srinivasan. The blocks of finite general linear and unitary groups. Invent. Math. , 69(1):109--153, 1982

    work page 1982

  11. [11]

    GAP -- Groups, Algorithms, and Programming, Version 4.15.1 , 2025

    The GAP Group. GAP -- Groups, Algorithms, and Programming, Version 4.15.1 , 2025. https://www.gap-system.org

    work page 2025

  12. [12]

    Geck and G

    M. Geck and G. Malle. The Character Theory of Finite Groups of Lie Type: A Guided Tour . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2020

    work page 2020

  13. [13]

    Gow and J

    R. Gow and J. Murray. Self-dual modules in characteristic two and normal subgroups. J. Algebra , 570:119--139, 2021

    work page 2021

  14. [14]

    Gow and W

    R. Gow and W. Willems. Quadratic geometries, projective modules, and idempotents. J. Algebra , 160(1):257--272, 1993

    work page 1993

  15. [15]

    Gow and W

    R. Gow and W. Willems. Methods to decide if simple self-dual modules over fields of characteristic 2 are of quadratic type. J. Algebra , 175(3):1067--1081, 1995

    work page 1995

  16. [16]

    Gow and W

    R. Gow and W. Willems. On the quadratic type of some simple self-dual modules over fields of characteristic two. J. Algebra , 195(2):634--649, 1997

    work page 1997

  17. [17]

    Hiss and J

    G. Hiss and J. Shamash. 2 -blocks and 2 -modular characters of the C hevalley groups G _2(q) . Math. Comp. , 59(200):645--672, 1992

    work page 1992

  18. [18]

    Kessar and G

    R. Kessar and G. Malle. Quasi-isolated blocks and B rauer's height zero conjecture. Ann. of Math. , 178:321--384, 2013

    work page 2013

  19. [19]

    Linckelmann

    M. Linckelmann. The Block Theory of Finite Group Algebras , volume 2 of London Mathematical Society Student Texts . Cambridge University Press, 2018

    work page 2018

  20. [20]

    L \"u beck

    F. L \"u beck. Data for finite groups of L ie type and related algebraic groups. https://www.math.rwth-aachen.de/ Frank.Luebeck/chev/index.html?LANG=en

    work page

  21. [21]

    G. Lusztig. On the representations of reductive groups with disconnected centre. Ast\' e risque , 168:157--166, 1988

    work page 1988

  22. [22]

    Malle and D

    G. Malle and D. Testerman. Linear Algebraic Groups and Finite Groups of Lie Type . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2011

    work page 2011

  23. [23]

    J. C. Murray and G. Navarro. Characters, bilinear forms and solvable groups. J. Algebra , 449:346--354, 2016

    work page 2016

  24. [24]

    Nagao and Y

    H. Nagao and Y. Tsushima. Representations of Finite Groups . Academic Press, San Diego, 1989

    work page 1989

  25. [25]

    G. Navarro. Characters and blocks of finite groups , volume 250 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge, 1998

    work page 1998

  26. [26]

    Navarro and P

    G. Navarro and P. H. Tiep. Characters of relative p' -degree over normal subgroups. Ann. of Math. , 178:1135--1171, 2013

    work page 2013

  27. [27]

    A. A. Schaeffer Fry, Taylor, and C. R. Vinroot. Galois automorphisms and a unique J ordan decomposition in the case of connected centralizer. J. Algebra , 664:123--149, 2025

    work page 2025

  28. [28]

    Singh and M

    A. Singh and M. Thakur. Reality properties of conjugacy classes in algebraic groups. Israel J. Math. , 165:1--27, 2008

    work page 2008

  29. [29]

    Srinivasan and C

    B. Srinivasan and C. R. Vinroot. Jordan decomposition and real-valued characters of finite reductive groups with connected center. Bull. Lond. Math. Soc. , 47(3):427--435, 2015

    work page 2015

  30. [30]

    Srinivasan and C

    B. Srinivasan and C. R. Vinroot. Galois group action and J ordan decomposition of characters of finite reductive groups with connected center. J. Algebra , 558:708--727, 2020

    work page 2020