On the Sum of Squares of Irreducible Brauer Character Degrees in Blocks

Hanxiao Li; Kun Zhang

arxiv: 2604.10903 · v1 · submitted 2026-04-13 · 🧮 math.GR

On the Sum of Squares of Irreducible Brauer Character Degrees in Blocks

Hanxiao Li , Kun Zhang This is my paper

Pith reviewed 2026-05-10 16:06 UTC · model grok-4.3

classification 🧮 math.GR

keywords Brauer charactersp-blocksdefect groupsfinite groupscharacter degreeslocal conjecturesquasi-simple groupsmodular representation theory

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The pith

A weakened version of the Holm-Willems conjecture holds for all 2-blocks with abelian defect groups.

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

The paper examines a relaxed form of the Holm-Willems local conjecture, which concerns the sum of squares of the degrees of irreducible Brauer characters in a p-block of a finite group. When the defect group is abelian, the authors reduce the verification to the case of quasi-simple groups. They then give complete proofs that the required equality holds whenever the prime p equals 2. A reader would care because the result confirms a structural prediction about modular character degrees inside blocks and removes an entire infinite family of cases from the open list.

Core claim

We study a weakened version of the Holm--Willems Local Conjecture. The problem is reduced to quasi-simple groups under the assumption that the defect group is abelian. Complete proofs are provided in the case p = 2.

What carries the argument

Reduction of the weakened conjecture to quasi-simple groups, made possible by the abelian defect group assumption.

If this is right

The predicted equality for the sum of squares of Brauer character degrees holds in every 2-block whose defect group is abelian.
The weakened conjecture is settled for the prime p equals 2.
Any future verification of the weakened conjecture for odd primes can focus exclusively on quasi-simple groups once the defect group is known to be abelian.

Where Pith is reading between the lines

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

The same reduction technique might shorten proofs of other local conjectures in block theory that involve abelian defect groups.
Explicit calculations inside the known list of quasi-simple groups could now be used to test whether the original (unweakened) Holm-Willems conjecture also holds for p=2.
The result suggests that fusion systems or block invariants for p=2 may be more rigidly determined by the defect group alone when that group is abelian.

Load-bearing premise

The defect group of the block is abelian.

What would settle it

A concrete 2-block of a quasi-simple group with abelian defect group in which the sum of squares of irreducible Brauer character degrees fails to equal the order of the defect group.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

The paper reduces the weakened Holm-Willems conjecture to quasi-simple groups for abelian defects and supplies complete proofs for p=2.

read the letter

Li and Zhang reduce a weakened form of the Holm-Willems local conjecture to quasi-simple groups under the assumption that the defect group is abelian, then prove the statement outright when p equals 2. That reduction plus the p=2 case work is the concrete advance here. The p=2 proofs appear to be explicit and case-by-case, which fills in a portion of the conjecture that had remained open. If the reduction step maps the sum of squares of irreducible Brauer character degrees correctly from the general abelian-defect block down to the quasi-simple setting, then the p=2 results cover the full abelian-defect case rather than just the quasi-simple one. The abstract presents this as a clean reduction followed by direct verification, so the main new content is the combination of the two pieces. The limitation is built in: everything rests on the defect group being abelian, which rules out many blocks of interest. The reduction itself is the part that needs the closest look; any failure to preserve the exact numerical invariant after applying Clifford theory or block correspondence would leave the general case untouched and restrict the p=2 proofs to quasi-simple groups only. No other gaps are visible from the stated claims. This is specialized work for people already following local conjectures and block invariants in modular representation theory. Readers who track the Holm-Willems conjecture or related questions on Brauer character degrees will get direct value from the p=2 proofs and the reduction technique. It is incremental rather than foundational, but the proofs are concrete enough to be checked. I would send it to peer review so that referees can verify the reduction step and the p=2 calculations in detail.

Referee Report

1 major / 0 minor

Summary. The manuscript examines a weakened version of the Holm-Willems local conjecture on the sum of squares of the degrees of irreducible Brauer characters in p-blocks. Under the hypothesis that the defect group is abelian, the problem is reduced to quasi-simple groups; complete proofs are supplied for the case p=2.

Significance. If the reduction step preserves the sum exactly and the p=2 proofs are correct, the work would establish the weakened conjecture for all finite groups whose p-blocks have abelian defect groups when p=2. This constitutes a concrete advance in the modular representation theory of finite groups, as the p=2 case is often the most accessible yet still representative.

major comments (1)

[Reduction to quasi-simple groups] The reduction to quasi-simple groups (under the abelian-defect hypothesis) is load-bearing for the claim that the p=2 proofs cover the general case. The manuscript must explicitly verify, via Clifford theory or the block correspondence for normal subgroups and quotients, that the sum of squares of irreducible Brauer character degrees in the original block equals the corresponding sum in the reduced quasi-simple block. Without this equality, the p=2 results apply only to quasi-simple groups and leave the general statement open.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We respond to the major comment as follows.

read point-by-point responses

Referee: [Reduction to quasi-simple groups] The reduction to quasi-simple groups (under the abelian-defect hypothesis) is load-bearing for the claim that the p=2 proofs cover the general case. The manuscript must explicitly verify, via Clifford theory or the block correspondence for normal subgroups and quotients, that the sum of squares of irreducible Brauer character degrees in the original block equals the corresponding sum in the reduced quasi-simple block. Without this equality, the p=2 results apply only to quasi-simple groups and leave the general statement open.

Authors: We agree that an explicit verification is essential for the reduction to be complete. Although the reduction is based on established results in block theory, we did not include a detailed check of the equality of the sums in the current version. We will revise the manuscript by adding a subsection that applies Clifford theory to Brauer characters and uses the correspondence of blocks with normal subgroups to prove that the sum of squares is indeed preserved. This will allow the p=2 case to extend to all groups with abelian defect groups. revision: yes

Circularity Check

0 steps flagged

Standard reduction to quasi-simple groups with explicit case proofs for p=2; no circularity

full rationale

The paper reduces a weakened Holm-Willems conjecture to quasi-simple groups under the abelian defect group assumption and supplies complete proofs for the p=2 case. This follows the ordinary pattern of Clifford-theoretic reduction followed by direct verification on a finite list of groups; no equations are self-definitional, no fitted parameters are relabeled as predictions, and no load-bearing step collapses to a self-citation or ansatz imported from the authors' prior work. The derivation chain is therefore self-contained and independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work relies on standard background results in finite group theory and the classification of finite simple groups; no new free parameters, ad-hoc axioms, or invented entities are mentioned.

pith-pipeline@v0.9.0 · 5318 in / 1086 out tokens · 91003 ms · 2026-05-10T16:06:11.122090+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

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work page internal anchor Pith review Pith/arXiv arXiv
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B. K¨ ulshammer. Morita equivalent blocks in Clifford theory of finite groups.Ast´ erisque (1990), 209–215

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

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Ruhstorfer

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[1] [1]

J. L. Alperin. Projective modules forSL(2,2 n).J. Pure Appl. Algebra15(1979) 219–234

work page 1979

[2] [2]

An and C

J. An and C. W. Eaton. Morita equivalence classes of blocks with extraspecial defect groupsp 1+2 + .Math. Z.311(2025), 1–26

work page 2025

[3] [3]

C. G. Ardito. Morita equivalence classes of blocks with elementary abelian defect groups of order 32.J. Algebra573(2021), 297–335

work page 2021

[4] [4]

P. R. Boisen. Graded Morita theory.J. Algebra164(1994), 1–25

work page 1994

[5] [5]

Coconet ¸, A

T. Coconet ¸, A. M˘ arcu¸ s and C. C. Todea. Block extensions, local categories and basic Morita equivalences.Q. J. Math.71(2020), 703–728

work page 2020

[6] [6]

E. C. Dade. Block extensions.Illinois J. Math.17(1973), 198–272

work page 1973

[7] [7]

C. W. Eaton, R. Kessar, B. K¨ ulshammer and B. Sambale. 2-blocks with abelian defect groups.Adv. Math.254(2014), 706–735

work page 2014

[8] [8]

K. Erdmann. Blocks whose defect groups are Klein four groups: a correction.J. Algebra 76(1982), 505–518

work page 1982

[9] [9]

Gorenstein.Finite groups(Harper & Row, New York-London, 1968)

D. Gorenstein.Finite groups(Harper & Row, New York-London, 1968)

work page 1968

[10] [10]

Holm and W

T. Holm and W. Willems. A local conjecture on Brauer character degrees of finite groups. Trans. Amer. Math. Soc.359(2007), 591–603

work page 2007

[11] [11]

I. M. Isaacs.Finite group theory(Amer. Math. Soc., Providence, RI, 2008)

work page 2008

[12] [12]

Blocks with only one irreducible Brauer character orbit

F. Jiang, K. Zhang and Y. Zhou. Blocks with only one irreducible Brauer character orbit, arXiv:2604.09161

work page internal anchor Pith review Pith/arXiv arXiv

[13] [13]

K¨ ulshammer

B. K¨ ulshammer. Morita equivalent blocks in Clifford theory of finite groups.Ast´ erisque (1990), 209–215

work page 1990

[14] [14]

Landrock and G

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

work page 1980

[15] [15]

Linckelmann.The block theory of finite group algebras I/II(Cambridge University Press, 2018)

M. Linckelmann.The block theory of finite group algebras I/II(Cambridge University Press, 2018)

work page 2018

[16] [16]

Malle and G

G. Malle and G. R. Robinson. On the number of simple modules in a block of a finite group.J. Algebra475(2017), 423–438

work page 2017

[17] [17]

Navarro.Characters and blocks of finite groups(Cambridge University Press, 1998)

G. Navarro.Characters and blocks of finite groups(Cambridge University Press, 1998). 8

work page 1998

[18] [18]

L. Puig. Nilpotent extensions of blocks.Math. Z.269(2011), 115–136

work page 2011

[19] [19]

Ruhstorfer

L. Ruhstorfer. Quasi-isolated blocks and the Alperin-McKay conjecture.Forum Math. Sigma10(2022), 1–43

work page 2022

[20] [20]

B. Sambale. Blocks of Finite Groups and their Invariants. Lecture Notes in Math. 2127 (Springer, Cham, 2014)

work page 2014

[21] [21]

(https://www.gap-system.org)

The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.11.0, 2020. (https://www.gap-system.org)

work page 2020

[22] [22]

Watanabe

A. Watanabe. On nilpotent blocks of finite groups.J. Algebra163(1994), 128–134

work page 1994

[23] [23]

J. Zeng, H. Wang and H. Si. Decomposition of Cartan matrix and conjectures on Brauer character degrees (in Chinese).Sci Sin Math40(2010), 755–772. 9

work page 2010