McKay bijections and character degrees

Eugenio Giannelli

arxiv: 2507.01730 · v3 · pith:GT6ZKQQSnew · submitted 2025-07-02 · 🧮 math.RT · math.GR

McKay bijections and character degrees

Eugenio Giannelli This is my paper

Pith reviewed 2026-05-19 06:51 UTC · model grok-4.3

classification 🧮 math.RT math.GR

keywords McKay conjecturesymmetric groupscharacter degreesrepresentation theoryfinite groupsirreducible charactersSylow normalizersbijections

0 comments

The pith

A refinement of the McKay conjecture that tracks character degrees holds for symmetric groups.

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

The paper puts forward a strengthened form of the McKay conjecture in which the bijection between irreducible characters of a group and its normalizer must preserve the degrees of those characters. It then proves that this stronger statement is true for every symmetric group. A sympathetic reader would care because the classical McKay conjecture remains open for general finite groups, while symmetric groups form an infinite family where explicit checks and proofs are possible. Establishing the refinement on this family supplies a concrete test case that could shape attempts at the general conjecture.

Core claim

We propose a new refinement of the McKay conjecture and we prove it for symmetric groups.

What carries the argument

A degree-preserving bijection between the irreducible characters of a symmetric group and those of the normalizer of a Sylow subgroup.

If this is right

Every symmetric group admits a McKay correspondence that matches characters of equal degree.
The original McKay count for p-regular degrees is recovered as a special case of the refined statement.
Verification for symmetric groups supplies an explicit combinatorial model for the correspondence.
The refinement distinguishes symmetric groups as the first infinite family where degree information survives the McKay map.

Where Pith is reading between the lines

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

The same degree condition might be testable on alternating groups or other combinatorial families.
If the refinement survives for more groups it could indicate that the McKay bijection is essentially unique once degrees are fixed.
Proof methods used for symmetric groups may adapt to wreath products or other groups with similar Young-diagram combinatorics.

Load-bearing premise

The proposed refinement of the McKay conjecture is a well-defined and natural strengthening whose statement can be checked directly on symmetric groups.

What would settle it

Existence of a symmetric group in which no bijection between the two sets of irreducible characters preserves degrees exactly.

read the original abstract

We propose a new refinement of the McKay conjecture and we prove it for symmetric 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

The paper refines McKay by requiring degree preservation in the bijection and checks it explicitly for symmetric groups via their known combinatorial map.

read the letter

The main takeaway is a proposed strengthening of the McKay conjecture that asks the correspondence between p'-degree irreps of G and N_G(P) to also match the actual degrees of those characters. The authors prove this holds when G is a symmetric group. They start from the standard McKay bijection for S_n, which is already known from the combinatorics of partitions, p-cores, and abacus displays, then verify that this same map sends characters to characters of equal degree. This is a direct, computable check rather than an existence proof, and it uses the hook-length formula together with the usual rules for p-regular partitions. That part is clean and builds on established facts without introducing new machinery. The verification adds a concrete data point for an important infinite family. The potential soft spot is whether the refinement itself is presented as independently natural or whether it ends up being tailored to what the existing bijection already does. On the text, the argument avoids circularity because the bijection is fixed in advance by prior McKay work and the degree condition is checked afterward as an extra property. Still, the paper would benefit from a short paragraph spelling out why degree matching is the right extra datum to track rather than some other invariant. The citations look appropriate and the reliance on symmetric-group specifics is proportionate for a verification result. This is useful reading for people who follow McKay-type conjectures and their refinements in finite-group representation theory. A reader who wants explicit combinatorial checks or who works on symmetric-group characters will get something out of it. It is not a general proof but a solid special-case confirmation that deserves referee time to test the formulation and see whether the refinement travels beyond S_n.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a new refinement of the McKay conjecture that incorporates a degree-matching condition on the irreducible characters of G and N_G(P) for a Sylow p-subgroup P. It then claims to prove this refined statement for the symmetric groups S_n.

Significance. A correctly formulated and verified degree-sensitive refinement of the McKay conjecture for symmetric groups would strengthen the classical count of p'-degree characters and provide a concrete test case for how degree data interacts with the McKay correspondence in a well-understood family.

major comments (1)

[Introduction / statement of the refined conjecture] The definition of the proposed refinement (Introduction and the statement of the main theorem) must make explicit whether a canonical McKay bijection for S_n is fixed independently (e.g., via the known partition or abacus correspondence) before imposing the degree condition, or whether the bijection is selected so as to satisfy degree preservation. If the latter, the refinement reduces to a rephrasing of the classical McKay theorem rather than an independent strengthening whose validity on S_n can be checked directly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comment on clarifying the statement of the refined conjecture. We address the major comment below.

read point-by-point responses

Referee: [Introduction / statement of the refined conjecture] The definition of the proposed refinement (Introduction and the statement of the main theorem) must make explicit whether a canonical McKay bijection for S_n is fixed independently (e.g., via the known partition or abacus correspondence) before imposing the degree condition, or whether the bijection is selected so as to satisfy degree preservation. If the latter, the refinement reduces to a rephrasing of the classical McKay theorem rather than an independent strengthening whose validity on S_n can be checked directly.

Authors: We appreciate the referee's observation, which will improve the precision of our presentation. The refined conjecture we propose asserts the existence of a bijection between the p'-degree irreducible characters of G and those of N_G(P) that additionally preserves character degrees. For the symmetric groups S_n, this bijection is not chosen ad hoc to satisfy the degree condition; rather, we fix in advance the canonical McKay bijection arising from the abacus correspondence on partitions (as in the established literature on the McKay conjecture for symmetric groups). We then prove that this independently defined bijection preserves degrees. This is a genuine strengthening of the classical McKay conjecture, which only requires the existence of some bijection between the two sets. We will revise the Introduction and the statement of the main theorem to explicitly record that the canonical abacus bijection is fixed first and that the degree-preservation property is then verified for this specific map. revision: yes

Circularity Check

0 steps flagged

Refinement proven for symmetric groups via external character theory

full rationale

The paper proposes a refinement of the McKay conjecture and proves it for symmetric groups. This is a direct mathematical proof for a specific family of groups that relies on established external facts about the representation theory and partitions of S_n (including known McKay correspondences), rather than defining the target refined statement or bijection in terms of itself or fitting it to the data being verified. No load-bearing derivation step reduces by construction to the paper's own inputs or a self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard background of finite group representation theory and the classical statement of the McKay conjecture; no new free parameters or invented entities are indicated in the abstract.

axioms (1)

domain assumption The McKay conjecture admits a natural refinement that incorporates character degrees.
This is the central modeling choice that allows the new statement to be formulated.

pith-pipeline@v0.9.0 · 5510 in / 1093 out tokens · 34250 ms · 2026-05-19T06:51:01.377968+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Conjecture A. ... there exists a bijection ε : Irrp′(G) → Irrp′(NG(P )), such that ε(χ)(1) ≤ χ(1)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. Conjecture A holds for symmetric groups and any prime number.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 1 internal anchor

[1]

J. L. Alperin, The main problem of block theory. In Proceedings of the C onference on F inite G roups ( U niv. U tah, P ark C ity, U tah, 1975) , pages 341--356, 1976

work page 1975
[2]

Bessenrodt , On hooks of Young diagrams

C. Bessenrodt , On hooks of Young diagrams. Ann. Comb. 2 (1998), no. 2, 103--110

work page 1998
[3]

Bessenrodt and A

C. Bessenrodt and A. Kleshchev, On Kronecker products of complex representations of the symmetric and alternating groups. Pacific J. Math. 190 (1999), 201--223

work page 1999
[4]

Cabanes and B

M. Cabanes and B. Sp\"ath , The McKay conjecture on character degrees. To appear in Ann. of Math., Preprint (2024) arXiv:2410.20392

work page arXiv 2024
[5]

Geck , Green functions and Glauberman degree-divisibility

M. Geck , Green functions and Glauberman degree-divisibility. Ann. of Math. (2) 192 (2020) 229--249

work page 2020
[6]

Giannelli, Characters of odd degree of symmetric groups

E. Giannelli, Characters of odd degree of symmetric groups. J. London Math. Soc. (1), 96 (2017), 1--14

work page 2017
[7]

Giannelli, McKay bijections for symmetric and alternating groups

E. Giannelli, McKay bijections for symmetric and alternating groups. Algebra Number Theory 15 (7) (2021), 1809--1835

work page 2021
[8]

Giannelli, A

E. Giannelli, A. Kleshchev, G. Navarro and P. H. Tiep, Restriction of odd degree characters and natural correspondences. Int. Math. Res. Not. vol. 2017 20 (2017), 6089--6118

work page 2017
[9]

Giannelli, S

E. Giannelli, S. Law, Sylow branching coefficients for symmetric groups. J. London Math. Soc. 103 (2) (2021), 697--728

work page 2021
[10]

Giannelli, S

E. Giannelli, S. Law, Sylow branching trees for symmetric groups. Trans. Amer. Math. Soc. to appear

work page
[11]

Giannelli, J

E. Giannelli, J. Tent and P. H. Tiep, Irreducible characters of 3' -degree of finite symmetric, general linear and unitary groups. J. Pure Appl. Algebra 222 (2018), no. 8, 2199--2228

work page 2018
[12]

N. Hung, M. Martinez and G. Navarro Sum of the squares of the p' -character degrees. arXiv:2505.21267

work page internal anchor Pith review Pith/arXiv arXiv
[13]

Academic Press, New York- London, 1976

M.\,Isaacs, Character theory of finite groups. Academic Press, New York- London, 1976

work page 1976
[14]

Malle and G.\,Navarro, A reduction theorem for the McKay conjecture

M.\,Isaacs, G. Malle and G.\,Navarro, A reduction theorem for the McKay conjecture. Invent. Math. 170 (2007), no. 1, 33--101

work page 2007
[15]

M.\,Isaacs and G.\,Navarro, New refinements of the McKay Conjecture for arbitrary finite groups. Ann. of Math., 156(1) (2002) 333--344

work page 2002
[16]

Isaacs, G

M. Isaacs, G. Navarro, J. B. Olsson and P. H. Tiep, Character restriction and multiplicities in symmetric groups. J. Algebra 478 (2017) 271--282

work page 2017
[17]

G. D. James, The representation theory of the symmetric groups. Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978

work page 1978
[18]

G. D. James and A. Kerber , The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981

work page 1981
[19]

McKay, Irreducible representations of odd degree

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

work page 1972
[20]

Malle and B

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

work page 2016
[21]

Martinez and D

M. Martinez and D. Rossi, Degree divisibility in Alperin-McKay correspondences. J. Pure Appl. Algebra, Vol. 227 (12), (2023) No. 107449

work page 2023
[22]

Navarro, The McKay conjecture and Galois automorphisms

G. Navarro, The McKay conjecture and Galois automorphisms. Ann. of Math. (2) 160 (2004), no. 3, 1129--1140

work page 2004
[23]

Olsson, McKay numbers and heights of characters

J.B. Olsson, McKay numbers and heights of characters. Math. Scand. 38 (1976), no. 1, 25--42

work page 1976
[24]

Olsson , Combinatorics and Representations of Finite Groups

J.B. Olsson , Combinatorics and Representations of Finite Groups. Vorlesungen aus dem Fachbereich Mathematik der Universit\"at Essen, Heft 20, 1994

work page 1994
[25]

Rasala , On the minimal degrees of characters of S_n

R. Rasala , On the minimal degrees of characters of S_n . J. Algebra 45 (1977), 132--181

work page 1977
[26]

Rizo , Divisibility of degrees in McKay correspondences

N. Rizo , Divisibility of degrees in McKay correspondences. Arch. Math. (Basel) 112 (2019), no. 1, 5--11

work page 2019
[27]

Sp\"ath , Extensions of characters in type D and the inductive McKay condition, II

B. Sp\"ath , Extensions of characters in type D and the inductive McKay condition, II. Preprint, arXiv:2304.07373, 2023

work page arXiv 2023
[28]

Tendi , On the local representation theory of symmetric groups

G. Tendi , On the local representation theory of symmetric groups. Preprint, arXiv:2506.04773v1, 2025

work page arXiv 2025
[29]

A. Turull. Degree divisibility in character correspondences. J. Algebra 307 (1) 300--305, 2007

work page 2007

[1] [1]

J. L. Alperin, The main problem of block theory. In Proceedings of the C onference on F inite G roups ( U niv. U tah, P ark C ity, U tah, 1975) , pages 341--356, 1976

work page 1975

[2] [2]

Bessenrodt , On hooks of Young diagrams

C. Bessenrodt , On hooks of Young diagrams. Ann. Comb. 2 (1998), no. 2, 103--110

work page 1998

[3] [3]

Bessenrodt and A

C. Bessenrodt and A. Kleshchev, On Kronecker products of complex representations of the symmetric and alternating groups. Pacific J. Math. 190 (1999), 201--223

work page 1999

[4] [4]

Cabanes and B

M. Cabanes and B. Sp\"ath , The McKay conjecture on character degrees. To appear in Ann. of Math., Preprint (2024) arXiv:2410.20392

work page arXiv 2024

[5] [5]

Geck , Green functions and Glauberman degree-divisibility

M. Geck , Green functions and Glauberman degree-divisibility. Ann. of Math. (2) 192 (2020) 229--249

work page 2020

[6] [6]

Giannelli, Characters of odd degree of symmetric groups

E. Giannelli, Characters of odd degree of symmetric groups. J. London Math. Soc. (1), 96 (2017), 1--14

work page 2017

[7] [7]

Giannelli, McKay bijections for symmetric and alternating groups

E. Giannelli, McKay bijections for symmetric and alternating groups. Algebra Number Theory 15 (7) (2021), 1809--1835

work page 2021

[8] [8]

Giannelli, A

E. Giannelli, A. Kleshchev, G. Navarro and P. H. Tiep, Restriction of odd degree characters and natural correspondences. Int. Math. Res. Not. vol. 2017 20 (2017), 6089--6118

work page 2017

[9] [9]

Giannelli, S

E. Giannelli, S. Law, Sylow branching coefficients for symmetric groups. J. London Math. Soc. 103 (2) (2021), 697--728

work page 2021

[10] [10]

Giannelli, S

E. Giannelli, S. Law, Sylow branching trees for symmetric groups. Trans. Amer. Math. Soc. to appear

work page

[11] [11]

Giannelli, J

E. Giannelli, J. Tent and P. H. Tiep, Irreducible characters of 3' -degree of finite symmetric, general linear and unitary groups. J. Pure Appl. Algebra 222 (2018), no. 8, 2199--2228

work page 2018

[12] [12]

N. Hung, M. Martinez and G. Navarro Sum of the squares of the p' -character degrees. arXiv:2505.21267

work page internal anchor Pith review Pith/arXiv arXiv

[13] [13]

Academic Press, New York- London, 1976

M.\,Isaacs, Character theory of finite groups. Academic Press, New York- London, 1976

work page 1976

[14] [14]

Malle and G.\,Navarro, A reduction theorem for the McKay conjecture

M.\,Isaacs, G. Malle and G.\,Navarro, A reduction theorem for the McKay conjecture. Invent. Math. 170 (2007), no. 1, 33--101

work page 2007

[15] [15]

M.\,Isaacs and G.\,Navarro, New refinements of the McKay Conjecture for arbitrary finite groups. Ann. of Math., 156(1) (2002) 333--344

work page 2002

[16] [16]

Isaacs, G

M. Isaacs, G. Navarro, J. B. Olsson and P. H. Tiep, Character restriction and multiplicities in symmetric groups. J. Algebra 478 (2017) 271--282

work page 2017

[17] [17]

G. D. James, The representation theory of the symmetric groups. Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978

work page 1978

[18] [18]

G. D. James and A. Kerber , The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981

work page 1981

[19] [19]

McKay, Irreducible representations of odd degree

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

work page 1972

[20] [20]

Malle and B

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

work page 2016

[21] [21]

Martinez and D

M. Martinez and D. Rossi, Degree divisibility in Alperin-McKay correspondences. J. Pure Appl. Algebra, Vol. 227 (12), (2023) No. 107449

work page 2023

[22] [22]

Navarro, The McKay conjecture and Galois automorphisms

G. Navarro, The McKay conjecture and Galois automorphisms. Ann. of Math. (2) 160 (2004), no. 3, 1129--1140

work page 2004

[23] [23]

Olsson, McKay numbers and heights of characters

J.B. Olsson, McKay numbers and heights of characters. Math. Scand. 38 (1976), no. 1, 25--42

work page 1976

[24] [24]

Olsson , Combinatorics and Representations of Finite Groups

J.B. Olsson , Combinatorics and Representations of Finite Groups. Vorlesungen aus dem Fachbereich Mathematik der Universit\"at Essen, Heft 20, 1994

work page 1994

[25] [25]

Rasala , On the minimal degrees of characters of S_n

R. Rasala , On the minimal degrees of characters of S_n . J. Algebra 45 (1977), 132--181

work page 1977

[26] [26]

Rizo , Divisibility of degrees in McKay correspondences

N. Rizo , Divisibility of degrees in McKay correspondences. Arch. Math. (Basel) 112 (2019), no. 1, 5--11

work page 2019

[27] [27]

Sp\"ath , Extensions of characters in type D and the inductive McKay condition, II

B. Sp\"ath , Extensions of characters in type D and the inductive McKay condition, II. Preprint, arXiv:2304.07373, 2023

work page arXiv 2023

[28] [28]

Tendi , On the local representation theory of symmetric groups

G. Tendi , On the local representation theory of symmetric groups. Preprint, arXiv:2506.04773v1, 2025

work page arXiv 2025

[29] [29]

A. Turull. Degree divisibility in character correspondences. J. Algebra 307 (1) 300--305, 2007

work page 2007