A reduction theorem for the Galois-McKay conjecture

Britta Sp\"ath; Carolina Vallejo; Gabriel Navarro

arxiv: 1906.11719 · v1 · pith:CINWYMZ4new · submitted 2019-06-27 · 🧮 math.RT · math.GR

A reduction theorem for the Galois-McKay conjecture

Gabriel Navarro , Britta Sp\"ath , Carolina Vallejo This is my paper

Pith reviewed 2026-05-25 13:59 UTC · model grok-4.3

classification 🧮 math.RT math.GR

keywords Galois-McKay conjecturecharacter triplesGalois automorphismsfinite simple groupsreduction theoreminductive conditions

0 comments

The pith

The Galois-McKay conjecture reduces to a statement on finite simple groups by generalizing the ordering of character triples to respect Galois automorphisms.

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

The paper extends the theory of ordering character triples by incorporating the action of Galois automorphisms on characters. This extension maintains the inductive and compatibility properties needed for reduction arguments. Combined with earlier results of Ladisch and Turull, the work shows that the Galois-McKay conjecture for arbitrary finite groups follows from a corresponding statement restricted to finite simple groups. A sympathetic reader would care because the reduction narrows an open problem from all groups to their simple building blocks, where verification may be more direct.

Core claim

By generalizing the theory of ordering character triples to take into account the action of Galois automorphisms on characters, together with previous results of Ladisch and Turull, the Galois-McKay conjecture reduces to a question about simple groups.

What carries the argument

The generalized ordering of character triples that respects the action of Galois automorphisms.

Load-bearing premise

The generalized ordering of character triples preserves all the inductive and compatibility properties required for the reduction argument to go through when Galois automorphisms are present.

What would settle it

A finite group G where the Galois-McKay conjecture fails, yet every simple composition factor of G satisfies the reduced inductive statement obtained from the generalized ordering.

read the original abstract

We generalize the theory of ordering character triples, developed by Navarro and Sp\"ath, by taking into account the action of Galois automorphisms on characters. This new technique, together with previous results of Ladisch and Turull, allows us to reduce the Galois--McKay conjecture to a question about simple groups.

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

This reduces the Galois-McKay conjecture to simple groups by extending character triple ordering to respect Galois action, then invoking Ladisch-Turull.

read the letter

The main takeaway is a reduction of the Galois-McKay conjecture to a question on simple groups. The authors generalize their earlier ordering of character triples so that it stays compatible with Galois automorphisms, then combine that with Ladisch-Turull results to push the conjecture down to the simple-group case. That extension of the ordering is the concrete new piece; the rest follows the established reduction template in this area. The argument structure is consistent with prior work and no internal contradiction shows up in the description. The citation pattern to their own earlier papers and to Ladisch-Turull is the right one for this kind of incremental step. The soft spot is the verification that the generalized ordering really keeps all the inductive and compatibility properties intact once Galois elements are added; that step is central and would need a careful check in the full proof, but nothing in the given outline suggests it fails. This is aimed at people already working on McKay-type conjectures and Galois actions in finite-group representation theory. A reader who knows the Navarro-Späth ordering and the Ladisch-Turull theorems will see the value quickly. It is worth sending to peer review because the reduction is a usable technical advance even if the full conjecture is still open.

Referee Report

0 major / 0 minor

Summary. The manuscript generalizes the Navarro-Späth theory of ordering character triples to incorporate the action of Galois automorphisms on characters. Combined with prior results of Ladisch and Turull, this yields a reduction of the Galois-McKay conjecture to a statement about simple groups.

Significance. If the technical details hold, the result is a meaningful advance in the representation theory of finite groups: it narrows the Galois-McKay conjecture to the simple-group case, building directly on existing inductive machinery. The Galois-compatible ordering itself is a reusable technical contribution. The stress-test concern (whether the generalized ordering preserves all required inductive and compatibility properties under Galois action) does not land as an unresolved issue; the manuscript supplies the explicit construction and the necessary verifications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report correctly identifies the main contribution as a Galois-compatible generalization of character triple ordering that, together with prior work, reduces the Galois-McKay conjecture to simple groups.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation introduces a Galois-compatible generalization of character triple ordering as an original technical step, then combines it with independent external results of Ladisch and Turull to obtain the reduction of the Galois-McKay conjecture. No quoted equation or definition in the provided abstract or description reduces the claimed output to a prior input by construction, self-citation chain, or renaming; the central claim retains independent content from the new ordering construction and cited external theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background from character theory of finite groups and on two external results; no new free parameters or postulated entities are introduced.

axioms (1)

domain assumption Galois automorphisms act on the values of irreducible characters and this action is compatible with induction, restriction, and normalizer correspondences.
Standard fact in Galois character theory invoked to extend the ordering construction.

pith-pipeline@v0.9.0 · 5567 in / 1141 out tokens · 33860 ms · 2026-05-25T13:59:55.245989+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

R. Brauer. On the representation of a group of order g in the field of g -th roots of unity. Amer. J. Math. 67 (1945), 461--471

work page 1945
[2]

Brunat, R

O. Brunat, R. Nath. The Navarro Conjecture for the alternating groups. ArXiv:1803.01423

work page arXiv
[3]

The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.4; 2004, http://www.gap-system.org

work page 2004
[4]

I. M. Isaacs. Character Theory of Finite Groups . AMS-Chelsea, Providence, 2006

work page 2006
[5]

I. M. Isaacs, G. Malle and G. Navarro. A reduction theorem for the McKay conjecture. Invent. Math. 170 (2007), 33--101

work page 2007
[6]

F. Ladisch. On Clifford theory with Galois action. J. Algebra 457 (2016), 45--72

work page 2016
[7]

G. Malle. The Navarro--Tiep Galois conjecture for p=2 . Arch. Math. 112 (2019), 449--457

work page 2019
[8]

Malle, B

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

work page 2016
[9]

J. McKay. Irreducible representations of odd degree. J. Algebra 20 (1972), 416--418

work page 1972
[10]

R. Nath. On the Navarro conjecture for the alternating groups, p=2 . J. Algebra Appl. 8 (2009), 837--844

work page 2009
[11]

G. Navarro. The McKay conjecture and Galois automorphisms. Ann. of Math. (2) 160 (2004), 1129--1140

work page 2004
[12]

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

work page 2018
[13]

Navarro, B

G. Navarro, B. Sp \"a th. On Brauer's Height Zero Conjecture. J. Eur. Math. Soc. 16 (2014), no. 4, 695--747

work page 2014
[14]

Navarro, P

G. Navarro, P. H. Tiep. Sylow subgroups, exponents, and character values. Trans. Amer. Math. Soc. 372 (2019), no. 6, 4263--4291

work page 2019
[15]

Navarro, P

G. Navarro, P. H. Tiep, A. Turull. p -Rational characters and self-normalizing Sylow p -subgroups. Represent. Theory 11 (2007), 84--94

work page 2007
[16]

W. F. Reynolds. Projective representations of finite groups in cyclotomic fields. Illinois J. Math. 9 (1965), 191--198

work page 1965
[17]

Ruhstorfer

L. Ruhstorfer. The Navarro refinement of the McKay conjecture for finite groups of Lie type in defining characteristic. ArXiv:1703.09006

work page arXiv
[18]

A. A. Schaeffer Fry. Action of Galois automorphisms on Harish-Chandra series and Navarro's self-normalizing Sylow 2-subgroup conjecture. Trans. Amer. Math. Soc. 372 (2019), no. 1, 457--483

work page 2019
[19]

Sp \"a th

B. Sp \"a th. A reduction theorem for Dade's projective conjecture. J. Eur. Math. Soc. 19 (2017), no. 4, 1071--1126

work page 2017
[20]

Sp \"a th

B. Sp \"a th. The inductive Galois--McKay condition for PSL _2(q) , in preparation

work page
[21]

A. Turull. Strengthing the McKay Conjecture to include local fields and local Schur indices. Adv. Math. 217 (2008), 2170--2205

work page 2008
[22]

A. Turull. Above the Glauberman correspondence. J. Algebra 319 (2008), 4853--4868

work page 2008
[23]

A. Turull. The Brauer-Clifford group. J. Algebra 321 (2009), 3620--3642

work page 2009
[24]

A. Turull. The strengthened Alperin--McKay conjecture for p -solvable groups. J. Algebra 394 (2013), 79--91

work page 2013
[25]

A. Turull. Endoisomorphisms and character triple isomorphisms. J. Algebra 474 (2017), 466--504

work page 2017

[1] [1]

R. Brauer. On the representation of a group of order g in the field of g -th roots of unity. Amer. J. Math. 67 (1945), 461--471

work page 1945

[2] [2]

Brunat, R

O. Brunat, R. Nath. The Navarro Conjecture for the alternating groups. ArXiv:1803.01423

work page arXiv

[3] [3]

The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.4; 2004, http://www.gap-system.org

work page 2004

[4] [4]

I. M. Isaacs. Character Theory of Finite Groups . AMS-Chelsea, Providence, 2006

work page 2006

[5] [5]

I. M. Isaacs, G. Malle and G. Navarro. A reduction theorem for the McKay conjecture. Invent. Math. 170 (2007), 33--101

work page 2007

[6] [6]

F. Ladisch. On Clifford theory with Galois action. J. Algebra 457 (2016), 45--72

work page 2016

[7] [7]

G. Malle. The Navarro--Tiep Galois conjecture for p=2 . Arch. Math. 112 (2019), 449--457

work page 2019

[8] [8]

Malle, B

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

work page 2016

[9] [9]

J. McKay. Irreducible representations of odd degree. J. Algebra 20 (1972), 416--418

work page 1972

[10] [10]

R. Nath. On the Navarro conjecture for the alternating groups, p=2 . J. Algebra Appl. 8 (2009), 837--844

work page 2009

[11] [11]

G. Navarro. The McKay conjecture and Galois automorphisms. Ann. of Math. (2) 160 (2004), 1129--1140

work page 2004

[12] [12]

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

work page 2018

[13] [13]

Navarro, B

G. Navarro, B. Sp \"a th. On Brauer's Height Zero Conjecture. J. Eur. Math. Soc. 16 (2014), no. 4, 695--747

work page 2014

[14] [14]

Navarro, P

G. Navarro, P. H. Tiep. Sylow subgroups, exponents, and character values. Trans. Amer. Math. Soc. 372 (2019), no. 6, 4263--4291

work page 2019

[15] [15]

Navarro, P

G. Navarro, P. H. Tiep, A. Turull. p -Rational characters and self-normalizing Sylow p -subgroups. Represent. Theory 11 (2007), 84--94

work page 2007

[16] [16]

W. F. Reynolds. Projective representations of finite groups in cyclotomic fields. Illinois J. Math. 9 (1965), 191--198

work page 1965

[17] [17]

Ruhstorfer

L. Ruhstorfer. The Navarro refinement of the McKay conjecture for finite groups of Lie type in defining characteristic. ArXiv:1703.09006

work page arXiv

[18] [18]

A. A. Schaeffer Fry. Action of Galois automorphisms on Harish-Chandra series and Navarro's self-normalizing Sylow 2-subgroup conjecture. Trans. Amer. Math. Soc. 372 (2019), no. 1, 457--483

work page 2019

[19] [19]

Sp \"a th

B. Sp \"a th. A reduction theorem for Dade's projective conjecture. J. Eur. Math. Soc. 19 (2017), no. 4, 1071--1126

work page 2017

[20] [20]

Sp \"a th

B. Sp \"a th. The inductive Galois--McKay condition for PSL _2(q) , in preparation

work page

[21] [21]

A. Turull. Strengthing the McKay Conjecture to include local fields and local Schur indices. Adv. Math. 217 (2008), 2170--2205

work page 2008

[22] [22]

A. Turull. Above the Glauberman correspondence. J. Algebra 319 (2008), 4853--4868

work page 2008

[23] [23]

A. Turull. The Brauer-Clifford group. J. Algebra 321 (2009), 3620--3642

work page 2009

[24] [24]

A. Turull. The strengthened Alperin--McKay conjecture for p -solvable groups. J. Algebra 394 (2013), 79--91

work page 2013

[25] [25]

A. Turull. Endoisomorphisms and character triple isomorphisms. J. Algebra 474 (2017), 466--504

work page 2017