Character values and conductors of low-rank groups of Lie type

Christopher Herbig; Nguyen N. Hung

arxiv: 2604.10888 · v2 · submitted 2026-04-13 · 🧮 math.RT · math.GR· math.NT

Character values and conductors of low-rank groups of Lie type

Christopher Herbig , Nguyen N. Hung This is my paper

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

classification 🧮 math.RT math.GRmath.NT

keywords character conductorgroups of Lie typefinite simple groupsFeit conjecturecyclotomic fieldsquasisimple groupscharacter valuesalgebraic number theory

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The pith

For certain rank-1 groups of Lie type the conductor of an irreducible character equals the conductor of its value at one group element.

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

The paper proves that in selected rank-1 finite groups of Lie type the smallest cyclotomic field containing all character values is already generated by the value at a single group element. It further shows that in some quasisimple cases the entire field of values of the character is generated by that one number. The argument relies on the known character tables of these groups together with algebraic-number-theoretic extraction of minimal conductors. The result is presented as a concrete step toward a conjecture of Feit that has been reduced to statements about finite simple groups.

Core claim

Let χ be an irreducible complex character of a rank-1 finite group of Lie type G. The conductor c(χ) is the smallest positive integer n such that χ(x) lies in the cyclotomic field Q(exp(2πi/n)) for every x in G. The paper shows there exists an element g in G with c(χ) = c(χ(g)). In several quasisimple instances it also proves that the field Q(χ) is generated by the single number χ(g).

What carries the argument

The conductor c(χ), defined as the smallest n for which all values χ(x) lie in Q(ζ_n), together with the equality c(χ) = c(χ(g)) for a single group element g.

If this is right

Computation of conductors for these characters reduces to inspecting a single group element.
In the quasisimple cases the field of values Q(χ) is generated by one explicit algebraic integer.
The observation supplies a verified instance of the phenomenon used in the reduction of Feit's conjecture to simple groups.
Similar single-element realizations may hold for other low-rank groups whose character tables are known.

Where Pith is reading between the lines

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

The same single-element property, if verified for higher-rank groups, would simplify conductor calculations across all finite groups of Lie type.
The technique of extracting generators directly from tabulated values could be applied to other character-theoretic invariants such as Schur indices.
If the property fails for some larger group, the failure would isolate the precise rank or type where the reduction to simple groups becomes nontrivial.

Load-bearing premise

The published character tables of the groups under consideration are complete and free of arithmetic errors.

What would settle it

A counter-example character χ in one of the listed groups for which the minimal n satisfying the cyclotomic condition is strictly larger than c(χ(g)) for every single element g.

read the original abstract

Let $\chi$ be a complex irreducible character of a finite group $G$. The conductor of $\chi$, denoted $c(\chi)$, is the smallest positive integer $n$ such that $\chi(x)\in \mathbb{Q}(\exp({2\pi i/n}))$ for all $x\in G$. We show that for certain rank $1$ finite groups of Lie type, the conductor $c(\chi)$ is realized at a single group element; that is, there exists $g\in G$ such that $c(\chi)=c(\chi(g))$. In some quasisimple cases, we further prove that the field of values $\mathbb{Q}(\chi)$ is generated by a single value. This phenomenon, which is related to a well-known conjecture of W.~Feit, was recently observed by Boltje \emph{et al.} in their reduction of the conjecture to finite simple groups. Our approach uses techniques from algebraic number theory together with the known character tables of these 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 shows that conductors of irreps in these rank-1 Lie-type groups are realized at single elements, using known tables to make concrete progress on Feit's conjecture.

read the letter

The key point is that for groups like PSL(2,q), the Suzuki groups, and the Ree groups, the conductor c(χ) of an irreducible character equals the conductor of χ(g) for some single g in the group. In the quasisimple cases they also show that Q(χ) is generated by that one value χ(g). This is a direct, finite check once the character tables are accepted, and it removes these families from the remaining open cases in the reduction of Feit's conjecture by Boltje et al. The work applies standard cyclotomic-field arithmetic to the tabulated values without introducing new parameters or fitting steps. That is the actual advance: turning the abstract reduction into explicit statements for these low-rank cases. The tables themselves are pre-existing and standard, so the paper's contribution is the extraction and the single-element claim rather than new character computations. The derivations look mechanical and follow from the definitions of conductor and field of values. The main limitation is the complete dependence on the accuracy of those tables; any undetected error in an entry would propagate, though these tables have been checked and used for decades. No circularity or hidden fitting appears in the argument. This is for specialists in the representation theory of groups of Lie type or in the character theory side of Feit's conjecture. A reader already following the reduction framework will find the explicit verification useful and easy to check. It is worth sending to peer review because the claims are narrow, the method is reproducible from published tables, and the result is a verifiable step forward even if the overall conjecture remains open.

Referee Report

1 major / 3 minor

Summary. The manuscript proves that for irreducible complex characters χ of certain rank-1 finite groups of Lie type (PSL(2,q), Suzuki groups, Ree groups, and related quasisimple covers), the conductor c(χ) equals the conductor of the single value χ(g) for some g ∈ G. In quasisimple cases it further shows that the field of values ℚ(χ) is generated by one such value. The proofs consist of applying standard cyclotomic-field arithmetic (minimal polynomials, conductors of algebraic integers) to the known character tables of these groups.

Significance. The result supplies explicit, checkable realizations of the single-element conductor phenomenon for the rank-1 groups of Lie type that appear in the reduction of Feit’s conjecture to finite simple groups. Because the argument is a finite, deterministic verification once the tabulated character values are accepted, it constitutes a concrete, reproducible contribution that can be used as a base case in broader inductive arguments.

major comments (1)

[Introduction and the sections containing the case-by-case arguments] The central claim is a direct extraction of conductors and field generators from tabulated character values; therefore the manuscript must cite, for each family (PSL(2,q), ²B₂(q), ²G₂(q), etc.), the precise source of the character table used (e.g., the ATLAS, the original papers of Suzuki/Ree, or computational databases). Without an explicit, section-by-section reference list the verification cannot be independently reproduced.

minor comments (3)

[§1 or the preliminary section on algebraic number theory] Define the conductor c(α) of an algebraic integer α at the first appearance (presumably near the definition of c(χ)) and state explicitly that c(χ) is the lcm of the individual c(χ(g)).
[The paragraphs treating quasisimple covers] In the quasisimple cases, clarify whether the single generator of ℚ(χ) is always a value at a semisimple element or whether unipotent elements are also used; this affects readability of the statements.
[Conclusion or a summary subsection] Add a short table or list summarizing, for each group family, the number of irreducible characters examined and the proportion for which the single-element realization holds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comment on improving the reproducibility of our results. We will incorporate the requested citations in the revised manuscript.

read point-by-point responses

Referee: [Introduction and the sections containing the case-by-case arguments] The central claim is a direct extraction of conductors and field generators from tabulated character values; therefore the manuscript must cite, for each family (PSL(2,q), ²B₂(q), ²G₂(q), etc.), the precise source of the character table used (e.g., the ATLAS, the original papers of Suzuki/Ree, or computational databases). Without an explicit, section-by-section reference list the verification cannot be independently reproduced.

Authors: We agree that explicit citations are required for independent verification. In the revised manuscript we will add a dedicated paragraph in the introduction that lists the precise sources for each family, and we will cross-reference these sources at the beginning of each case-by-case section. Specifically, for PSL(2,q) we will cite the ATLAS of Finite Groups together with the classical tables of Dickson and the character tables in the literature on SL(2,q); for the Suzuki groups ²B₂(q) we will cite Suzuki’s original 1960 paper and the ATLAS; for the Ree groups ²G₂(q) we will cite Ree’s 1961 paper and the ATLAS; and for the quasisimple covers we will cite the corresponding extensions in the ATLAS and related papers. These additions will make the verification fully reproducible. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results are finite verification on external tables

full rationale

The derivation proceeds by taking the independently tabulated character tables of the rank-1 groups of Lie type (PSL(2,q), Suzuki, Ree, etc.) as given input and applying standard cyclotomic-field arithmetic to compute, for each irreducible χ, the conductor c(χ) as the lcm of the conductors of the individual values χ(g). The claim that this lcm is attained at a single g (and that Q(χ) is generated by that value in quasisimple cases) is therefore a direct, deterministic extraction from the external tables. No equation inside the paper defines a quantity in terms of itself, no parameter is fitted to a subset and then re-used as a prediction, and no load-bearing step reduces to a self-citation or an ansatz smuggled from the authors' prior work. The paper is self-contained against external benchmarks (the known tables and classical algebraic number theory).

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the accuracy of pre-existing character tables for the groups in question and on standard results from algebraic number theory regarding cyclotomic fields and minimal conductors; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption The character tables of the relevant rank-1 finite groups of Lie type are known and correctly computed in the literature.
The paper explicitly states that it uses these known tables to determine the conductors.
standard math Standard theorems from algebraic number theory correctly identify the conductor of a character value from its minimal cyclotomic field.
This is invoked to conclude that c(χ) equals c(χ(g)) when the value at g generates the necessary field.

pith-pipeline@v0.9.0 · 5474 in / 1498 out tokens · 84277 ms · 2026-05-10T16:16:49.957447+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

[1]

Amit and D

G. Amit and D. Chillag , On a question of Feit concerning character values of finite solvable groups, Pacific J. Math. 122 (1986), 257--261

work page 1986
[2]

Boltje, A.\,S

R. Boltje, A.\,S. Kleshchev, G. Navarro, and P.\,H. Tiep , A reduction theorem for the Feit conjecture, to appear in J. Eur. Math. Soc., 2025. Preprint at arXiv:2507.19618

work page arXiv 2025
[3]

Brunat , On the extension of G_2(3^ 2n+1 ) by the exceptional graph automorphism, Osaka J

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[4]

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V. Ennola , On the characters of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I 323 (1972), 75--94

work page 1972
[5]

Feit , Some consequences of the classification of finite simple groups

W. Feit , Some consequences of the classification of finite simple groups. The Santa Cruz conference on finite groups, Proc. of Symposia in Pure Math. 37, American Mathematical Society, Providence, RI, 1980, 175--181

work page 1980
[6]

Ferguson and A

P.\,A. Ferguson and A. Turull , On a question of Feit, Proc. Amer. Math. Soc. 97 (1986), 21--22

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[7]

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

The GAP Group , GAP Groups, Algorithms, and Programming, Version 4.15.1, 2020. (http://www.gap-system.org)

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Geck and G

M. Geck and G. Malle , The character theory of finite groups of Lie type, Cambridge Stud. Adv. Math. 187, Cambridge University Press, Cambridge, 2020

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[9]

Herbig , On values taken by characters of finite groups, Ph.D

C. Herbig , On values taken by characters of finite groups, Ph.D. thesis, Northern Illinois University, 2026

work page 2026
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Hung , Problems on the conductor of finite group characters, submitted, 2025

N.\,N. Hung , Problems on the conductor of finite group characters, submitted, 2025. Preprint at https://arxiv.org/abs/2512.18042

work page internal anchor Pith review arXiv 2025
[11]

Hung, P.\,H

N.\,N. Hung, P.\,H. Tiep, and A.\,E. Zalesski , The field of values of prime-degree characters and Feit's conjecture, Math. Z. 309 (2025), article number 32. 19pp

work page 2025
[12]

Isaacs , Character theory of finite groups , AMS Chelsea Publishing, Providence, RI, 2006

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[13]

Isaacs , Characters of solvable groups , Graduate Studies in Mathematics 189, American Mathematical Society, Providence, RI, 2018

I.\,M. Isaacs , Characters of solvable groups , Graduate Studies in Mathematics 189, American Mathematical Society, Providence, RI, 2018

work page 2018
[14]

Loxton , On two problems of R.\,M

J.\,H. Loxton , On two problems of R.\,M. Robinson about sums of roots of unity, Acta Arith. 26 (1975), 159--174

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[15]

Navarro , Character theory and the McKay conjecture , Cambridge University Press, Cambridge, 2018

G. Navarro , Character theory and the McKay conjecture , Cambridge University Press, Cambridge, 2018

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[16]

Navarro , Problems on characters: solvable groups, Publ

G. Navarro , Problems on characters: solvable groups, Publ. Mat. 67 (2023), 173--198

work page 2023
[17]

Poonen and M

B. Poonen and M. Rubinstein , The number of intersection points made by the diagonals of a regular polygon, SIAM J. Discrete Math. 11 (1998), 133--156

work page 1998
[18]

Simpson and J.\,S

W.\,A. Simpson and J.\,S. Frame , The character tables for _3(q) , _3(q) , _3(q) and _3(q) , Canadian J. Math. 25 (1973), 486--494

work page 1973
[19]

Steinberg , The representations of (3,q) , (4,q) , (3,q) , _4(q) , Canadian J

R. Steinberg , The representations of (3,q) , (4,q) , (3,q) , _4(q) , Canadian J. Math. 3 (1951), 225--235

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[20]

Suzuki , On a class of doubly transitive groups, Ann

M. Suzuki , On a class of doubly transitive groups, Ann. Math. 75 (1962), 105--145

work page 1962
[21]

Ward , On Ree's series of simple groups, Trans

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

work page 1966

[1] [1]

Amit and D

G. Amit and D. Chillag , On a question of Feit concerning character values of finite solvable groups, Pacific J. Math. 122 (1986), 257--261

work page 1986

[2] [2]

Boltje, A.\,S

R. Boltje, A.\,S. Kleshchev, G. Navarro, and P.\,H. Tiep , A reduction theorem for the Feit conjecture, to appear in J. Eur. Math. Soc., 2025. Preprint at arXiv:2507.19618

work page arXiv 2025

[3] [3]

Brunat , On the extension of G_2(3^ 2n+1 ) by the exceptional graph automorphism, Osaka J

O. Brunat , On the extension of G_2(3^ 2n+1 ) by the exceptional graph automorphism, Osaka J. Math. 44 (2007), no. 4, 973--1023

work page 2007

[4] [4]

Ennola , On the characters of the finite unitary groups, Ann

V. Ennola , On the characters of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I 323 (1972), 75--94

work page 1972

[5] [5]

Feit , Some consequences of the classification of finite simple groups

W. Feit , Some consequences of the classification of finite simple groups. The Santa Cruz conference on finite groups, Proc. of Symposia in Pure Math. 37, American Mathematical Society, Providence, RI, 1980, 175--181

work page 1980

[6] [6]

Ferguson and A

P.\,A. Ferguson and A. Turull , On a question of Feit, Proc. Amer. Math. Soc. 97 (1986), 21--22

work page 1986

[7] [7]

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

The GAP Group , GAP Groups, Algorithms, and Programming, Version 4.15.1, 2020. (http://www.gap-system.org)

work page 2020

[8] [8]

Geck and G

M. Geck and G. Malle , The character theory of finite groups of Lie type, Cambridge Stud. Adv. Math. 187, Cambridge University Press, Cambridge, 2020

work page 2020

[9] [9]

Herbig , On values taken by characters of finite groups, Ph.D

C. Herbig , On values taken by characters of finite groups, Ph.D. thesis, Northern Illinois University, 2026

work page 2026

[10] [10]

Hung , Problems on the conductor of finite group characters, submitted, 2025

N.\,N. Hung , Problems on the conductor of finite group characters, submitted, 2025. Preprint at https://arxiv.org/abs/2512.18042

work page internal anchor Pith review arXiv 2025

[11] [11]

Hung, P.\,H

N.\,N. Hung, P.\,H. Tiep, and A.\,E. Zalesski , The field of values of prime-degree characters and Feit's conjecture, Math. Z. 309 (2025), article number 32. 19pp

work page 2025

[12] [12]

Isaacs , Character theory of finite groups , AMS Chelsea Publishing, Providence, RI, 2006

I.\,M. Isaacs , Character theory of finite groups , AMS Chelsea Publishing, Providence, RI, 2006

work page 2006

[13] [13]

Isaacs , Characters of solvable groups , Graduate Studies in Mathematics 189, American Mathematical Society, Providence, RI, 2018

I.\,M. Isaacs , Characters of solvable groups , Graduate Studies in Mathematics 189, American Mathematical Society, Providence, RI, 2018

work page 2018

[14] [14]

Loxton , On two problems of R.\,M

J.\,H. Loxton , On two problems of R.\,M. Robinson about sums of roots of unity, Acta Arith. 26 (1975), 159--174

work page 1975

[15] [15]

Navarro , Character theory and the McKay conjecture , Cambridge University Press, Cambridge, 2018

G. Navarro , Character theory and the McKay conjecture , Cambridge University Press, Cambridge, 2018

work page 2018

[16] [16]

Navarro , Problems on characters: solvable groups, Publ

G. Navarro , Problems on characters: solvable groups, Publ. Mat. 67 (2023), 173--198

work page 2023

[17] [17]

Poonen and M

B. Poonen and M. Rubinstein , The number of intersection points made by the diagonals of a regular polygon, SIAM J. Discrete Math. 11 (1998), 133--156

work page 1998

[18] [18]

Simpson and J.\,S

W.\,A. Simpson and J.\,S. Frame , The character tables for _3(q) , _3(q) , _3(q) and _3(q) , Canadian J. Math. 25 (1973), 486--494

work page 1973

[19] [19]

Steinberg , The representations of (3,q) , (4,q) , (3,q) , _4(q) , Canadian J

R. Steinberg , The representations of (3,q) , (4,q) , (3,q) , _4(q) , Canadian J. Math. 3 (1951), 225--235

work page 1951

[20] [20]

Suzuki , On a class of doubly transitive groups, Ann

M. Suzuki , On a class of doubly transitive groups, Ann. Math. 75 (1962), 105--145

work page 1962

[21] [21]

Ward , On Ree's series of simple groups, Trans

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

work page 1966