The Isaacs--Navarro Galois conjecture

A. A. Schaeffer Fry; L. Ruhstorfer

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The Isaacs--Navarro Galois conjecture

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Pith's one-line read This paper proves the Isaacs–Navarro Galois conjecture: for every finite group and prime ℓ, an H0-equivariant bijection exists between the ℓ′-degree irreducible characters and the characters of the normalizer of a Sylow ℓ-subgroup.

desk verdict Full proof of the Isaacs–Navarro Galois conjecture for all primes; solid Lie-type machinery, but the D4(q) MAGMA assertion and unrefereed companions need referee scrutiny. read the letter →

arxiv 2509.02300 v1 pith:EN2OJTWI submitted 2025-09-02 math.RT math.GR

L. Ruhstorfer , A. A. Schaeffer Fry This is my paper · ORCID

classification math.RTmath.GR MSC 20C1520C3320D20

keywords Isaacs–NavarroGaloisconjectureMcKay–NavarroactiononcharactersirreduciblegroupsofLietypeℓ-rationalitylocal–globalconjecturesSylowsubgroups

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves the Isaacs–Navarro Galois conjecture, the 2002 refinement of the McKay conjecture that tracks Galois fields of values alongside character degrees. The main theorem asserts that for any finite group G and prime ℓ, there is a bijection between the irreducible characters of ℓ′-degree in G and those in the normalizer of a Sylow ℓ-subgroup D, and this bijection commutes with the Galois automorphisms in H0. Since H0 controls which ℓ′-roots of unity appear in character values, the bijection preserves more than degrees: it matches fields of values up to the ℓ-part of their conductors. From this the paper derives consequences that had remained open for odd primes, including an equivalence between the exponent of D/D′ and the fixed points of certain Galois automorphisms, and a conjecture about the ℓ-rationality levels of ℓ′-degree characters. The reason to care is that these are local–global statements: data visible in the ordinary character table of G now determine structural facts about Sylow subgroups.

What carries the argument

The machinery is the H0-equivariant extension map assembled via a structural splitting of the normalizer of a Sylow d-torus centralizer. Lemma 6.1 asserts that for N-hat = N_{GE(G^F)}(L), there is a subgroup V-hat with N-hat = L V-hat and V-hat ∩ L a 2-group contained in Z(L), plus an ℓ′-subgroup E-hat inside V-hat with N-hat = N E-hat. This splitting lets Lemma 3.2 extend characters from L to their inertia groups while preserving their stabilizers under H0 and the relevant automorphisms. Around that core, the proof organizes the Lie-type case through the reduction criteria of [RSST25] and [NSV20], reducing the inductive condition to three manageable requirements: an equivariant extension ma

What would settle it

Exhibit a prime ℓ and a finite reductive group G^F, e.g. a D4(q) case or an exceptional family not covered by the direct argument, for which no subgroup V-hat of N-hat = N_{GE(G^F)}(L) satisfies N-hat = L V-hat with V-hat ∩ L a 2-group contained in Z(L) and an ℓ′-subgroup E-hat ≤ V-hat with N-hat = N E-hat. A computer search using the Steinberg presentation of the group would settle the existence. If such a group exists, Corollary 6.4's extension maps fail, so the inductive Isaacs–Navarro condition for that simple group would not follow.

Watch

Extended reading notes

Core claim

The central claim, Theorem A, is that the Isaacs–Navarro Galois conjecture holds for all finite groups and all primes ℓ. Explicitly, if D is a Sylow ℓ-subgroup of G, there is an H0-equivariant bijection Irr_{ℓ′}(G) → Irr_{ℓ′}(N_G(D)), where H0 is the Galois group of automorphisms of Q^ab that act trivially on ℓ′-roots of unity and have ℓ-power order. The proof uses the reduction of the Galois–McKay family to simple groups and verifies the resulting inductive Isaacs–Navarro condition. The bulk of the verification concerns finite groups of Lie type in non-defining characteristic, where the authors construct the required extension maps and stable transversals, treating type A and regular d-valu

Load-bearing premise

The Lie-type part of the proof depends on Lemma 6.1's structural splitting: in every case, the normalizer of the Sylow d-torus centralizer must factor as L times a small group whose intersection with L is a 2-group inside the center, with a complementary ℓ′-piece. If one group family fails that splitting, the extension maps that carry the H0-equivariant bijection cannot be constructed by this proof.

Editorial extensions

If this is right

If Theorem A is right, then for every finite group G and prime ℓ, the exponent of D/D′ is bounded by ℓ^e exactly when every ℓ′-degree character is fixed by the Galois automorphism σ_e, and this is equivalent to the same fixed-point statement for the principal ℓ-block (Corollary B).
If Theorem A is right, then ℓ′-degree characters with ℓ-rationality level e ≥ 2 force characters at every intermediate level, and the existence of such characters is invariant under passage to subgroups of ℓ′-index (Corollary C).
If Theorem A is right, the original Isaacs–Navarro Galois conjecture is settled uniformly for odd primes, not just the prime 2 case that was known via separate arguments.
If Theorem A is right, the inductive Isaacs–Navarro condition holds for every nonabelian simple group, so the reduction theorem of [NSV20] can be applied in full scope.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof does not by itself advance the stronger H_ℓ version of the McKay–Navarro conjecture; H0 cannot see the difference between ℓ-rationality levels 0 and 1, so the ℓ-rationality gap phenomenon remains outside its reach.
A direct application one could test now is algorithmic: in concrete small groups, use the corollary to recover exp(D/D′) from the character table, which would make the classical character-table detection question computable rather than existential.
The splitting lemma used here may transfer to other local–global problems: any future equivariant bijection for Lie-type groups at odd primes will likely need a similar decomposition of N-hat into L times a 2-group with an ℓ′-complement, so the lemma is a reusable structural tool.
One open direction the paper leaves implicit is whether the H0-equivariant bijections constructed here can be upgraded to H_ℓ-equivariant bijections family-by-family; the missing ingredient is control of rationality levels 0 versus 1, not the underlying extension maps.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is the real thing. The paper proves the Isaacs–Navarro Galois conjecture for every finite group and every prime, which is a major theorem in finite group representation theory. The structure is exactly what you'd hope: NSV20 reduction, then the heavyweight Lie-type work in Sections 5–7. The genuinely new content is the H0-equivariant extension maps and transversals for non-type-A groups, especially Lemmas 6.6, 6.7, 6.10, and Proposition 7.2, and the inductive argument in Theorem 7.4 that handles the non-regular case. The type-A case and the regular case fall out cleanly once that machinery is up. The paper is also honest about what it takes from [RSST25] and [CS25], and the internal logic is coherent as far as I can check.

The soft spots, in proportion. The proof leans on two to-appear or unrefereed companions, [RSST25] and [CS25]. That is common in this area and not a mathematical flaw, but a referee needs stable versions and a precise statement of which hypotheses of Theorem 5.1 come from where. The more concrete issue is Lemma 6.1. For G = D4(q) with d = 3 or 6, the authors write that a MAGMA calculation shows V̂3 is C3×C3, and no code or certificate is provided. The stress-test note is right that this is load-bearing: Corollary 6.2’s Sylow decomposition and Proposition 7.2’s extension maps for that D4(q) family depend on it. If the 3-structure were wrong, the proof would not cover that infinite family. I think the assertion is very likely true and checkable in an afternoon, but “a calculation in MAGMA” is not a proof. This is a minor-to-moderate gap, not a fatal one. The central argument holds up.

One point in favor worth naming: the paper does not oversell. It proves the H0 version, not the full Hℓ McKay–Navarro conjecture, and says so clearly. The corollaries are genuine consequences, and the discussion of what does not follow is a model of restraint.

My verdict is close to the reader’s conditional. I would send this to a serious referee. Before publication I would ask for either the MAGMA input or a short proof for the D4(q) claim, and for stable links to the companion papers. Desk rejection would be wrong.

Referee Report

2 major / 4 minor

Summary. The paper proves the Isaacs–Navarro Galois conjecture (Theorem A): for every finite group G and every prime ℓ, with D ∈ Syl_ℓ(G), there is an H_0-equivariant bijection Irr_{ℓ′}(G) → Irr_{ℓ′}(N_G(D)), where H_0 is the Galois group acting trivially on ℓ′-roots of unity. The proof follows the standard reduction of Navarro–Späth–Vallejo to a condition on simple groups, handles the non-Lie-type and defining-characteristic cases via prior results, and then treats groups of Lie type in non-defining characteristic. For Lie-type groups the authors use the criteria of Ruhstorfer–Schaeffer Fry–Späth–Taylor and construct the required extension maps and transversals, with a new extension lemma (Lemma 6.1) and an induction argument for the non-regular case (Theorem 7.4). Corollaries B and C are derived, giving the exponent-of-Sylow-abelianization statement and Hung’s p-rationality-level conjectures.

Significance. If the proof is correct, this settles a central conjecture from 2002 and provides substantial corollaries on Galois actions on character values, extending the completed McKay conjecture. The paper is well-organized, builds on the recent proof of the McKay conjecture by Cabanes–Späth and on the reduction work of Navarro–Späth–Vallejo and Ruhstorfer–Schaeffer Fry–Späth–Taylor, and it isolates the new Lie-type extension problems clearly. The internal logic is consistent: the induction on the order of G/Z(G) is legitimate, and the paper does not assume the target conjecture. The main weakness is that one load-bearing step, the D_4(q) case of Lemma 6.1, relies on an unreproduced MAGMA computation. Because that step supports several later results used for the non-regular inductive step, the manuscript needs to make this computation verifiable before the proof can be considered complete.

major comments (2)

[Section 6.1, Lemma 6.1, D4(q) case (page 9)] The proof of Lemma 6.1 for G = D4(q) with d ∈ {3,6} contains the sentence: “A calculation in MAGMA shows that V̂3 ≅ C3×C3.” No MAGMA code, input file, or output is supplied. This is not a peripheral remark: the asserted structure is used to choose a Sylow 3-twist v, then to construct V̂0 and Ê0. Corollary 6.2 derives from this the Sylow ℓ-decomposition D = Z(L)_ℓ ⋊ V0 and the equality N_Ĝ(D) = N_G(D)Ê; those consequences are subsequently used in Corollary 6.4, Lemma 6.7, Lemma 6.11, and Proposition 7.2. If the computational assertion is false, the extension maps and transversals for the infinite family D_4(q) with ℓ | q²±q+1 are not established. Please provide the MAGMA code and transcript, or a hand proof of the claim that a Sylow 3-subgroup of V0Γ is C3×C3.
[Section 7.2, proof of Theorem 7.4] The line “By our assumption and applying [NSV20, Thm. A], the Isaacs–Navarro Galois conjecture holds for the group W~λ” is compressed. The induction hypothesis covers simple groups involved in groups of order smaller than |G/Z(G)|. For the argument to be valid, the relative Weyl group W~λ = N_G~(L, λ~)/L~ must have all its composition factors within that class. Since W~λ is a section of N_G~(L)/L~, which is a subgroup of the Weyl group (up to the regular-embedding center), its order is bounded uniformly in q and is indeed smaller for all but possibly trivial small q. Please spell out this order comparison explicitly, as the induction step is the mechanism that completes the proof for all non-regular d.

minor comments (4)

[Introduction, Section 1] The Galois group and the finite group are both denoted G: “the Galois group G := Gal(Q^ab/Q) acts naturally on Irr(G).” This is confusing; use a different letter, e.g. 𝒢, for the Galois group.
[Section 8] Typo: “Isaccas–Navarro” should be “Isaacs–Navarro”.
[Section 6.1, Lemma 6.1] The notation V̂3 is introduced as “a Sylow 3-subgroup of V0Γ”, but the sentence “A calculation in MAGMA shows that V̂3 ≅ C3×C3” would benefit from stating explicitly which group (V0Γ or W0Γ) the computation concerns.
[Throughout] Some inline formulas are poorly broken across lines (e.g. “N G(D) = NG(D)Ê” and “L NG(D)”). Please improve typesetting for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A is derived via an external reduction theorem and independent structural lemmas; self-citations are used as prior conditional reductions, not as renamed versions of the target result.

full rationale

The proof of Theorem A is self-contained as a derivation: it starts from the NSV20 reduction (Theorem 2.1), an external theorem reducing the Isaacs–Navarro conjecture to the inductive Galois–McKay condition for simple groups, and then checks that condition. Section 4 dispatches non-Lie-type, defining-characteristic, and ℓ=2 cases using published results including the authors' own [RSF25], which is a separate, stronger theorem for ℓ=2 and is not the same as the present H0-statement. Section 5 imports a conditional reduction from [RSST25, Thm. B] and [RSST25, Cor. 3.5]; that reduction has its own hypotheses (existence of extension maps and transversals) and does not assume Theorem A. The paper then verifies those hypotheses in Sections 6–7. The induction in Theorem 7.4 is legitimate: it applies the induction hypothesis only to groups of smaller order and then invokes [NSV20, Thm. A], a standard reduction, to obtain the Isaacs–Navarro bijection for the smaller relative Weyl group; the size comparison between W~λ and G/Z(G) is implicit but standard, so this is at most an omitted detail, not circularity. The flagged MAGMA assertion in Lemma 6.1 ('A calculation in MAGMA shows that V̂3 ≅ C3×C3') is load-bearing for the D4(q) non-regular case and lacks a supplied certificate, but it is an independent computational check, not a step that assumes the desired bijection; this is a reproducibility concern, not a circularity. There are no fitted parameters, no predictions-from-fit, and no renaming of a known result as a new derivation. The self-citations to [RSST25] and [RSF25] are to separate works with their own hypotheses and are used as reductions or prior cases, so they do not make the main claim equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

Pure existence proof; no fitted numerical constants. The proof is a reduction chain: global conjecture to inductive simple-group condition, then to explicit structural checks for groups of Lie type. The novel contribution is the extension-map and transversal lemmas; most ingredients are imported from prior work, with the companion preprint [RSST25] and to-appear [CS25] carrying substantial weight.

assumptions (9)

domain assumption The inductive Galois-McKay reduction [NSV20, Thm. A] is valid with H0 in place of Hl.
Theorem 2.1 asserts the global conjecture reduces to the inductive condition for simple groups; this is cited, not proved.
domain assumption The structural and equivariant results of [RSST25], especially Theorem B and Corollary 3.5, are correct.
Theorems 5.1 and 7.4 rely on the equivariant bijection between Irr of the regular-embedding group and Irr of N-hat; this is an arXiv preprint involving two of the present authors, with assumptions that do not include Theorem A.
domain assumption The McKay conjecture and the inductive McKay condition hold for all finite groups, as established in [CS25, Thm. B].
Used in Lemma 6.7, Theorem 7.4, and in the initial reduction; [CS25] is to appear in Annals of Mathematics.
domain assumption Jordan decomposition of characters of finite reductive groups commutes with automorphisms and with the Galois group as described in [CS13, SV20, Ma07, Prop. 7.3].
Theorem 7.4 uses equivariance of the bijection over semisimple classes and relative Weyl groups; these are imported results.
domain assumption The Sylow d-torus and Tits extended Weyl group machinery developed by Spath ([Sp09], [Sp10a], [Sp10b], [Sp23a], [Sp24]) is correct.
Notation 5.1 and Lemma 6.1 are built on these structural results for Sylow d-tori and their normalizers.
domain assumption Simple groups outside the generic Lie-type case satisfy the inductive conditions: alternating, sporadic, exceptional multiplier, Suzuki/Ree, and l=2 cases, as cited in Lemmas 4.1 and 4.2.
The proof of Theorem A depends on these prior results to reduce to groups of Lie type in non-defining characteristic.
ad hoc to paper A MAGMA computation asserts that a Sylow 3-subgroup of V0Gamma for D4(q) is C3 x C3.
Lemma 6.1 uses this to produce a Sylow 3-twist; the script and output are not included.
standard math Standard character theory results: Gallagher's lemma, Clifford theory, determinantal orders, and extension uniqueness as in [Isa76].
Used throughout Section 3, e.g., in Lemma 3.2 and the proof of Theorem 5.1.
domain assumption The classification of finite simple groups and the description of universal covering groups as G^F with generic Schur multiplier for the remaining cases.
Section 5 reduces the problem to quasisimple groups of Lie type with generic Sylow normalizer structure in the sense of [Ma07, Thm. 5.14].

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Pith. "Pith review of The Isaacs--Navarro Galois conjecture." pith.science (2026). https://pith.science/paper/EN2OJTWI

@misc{pith2026250902300,
  author       = {Pith},
  title        = {Pith review of: The Isaacs--Navarro Galois conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EN2OJTWI}},
  note         = {Machine review of arXiv:2509.02300}
}

read the original abstract

We prove the original Galois refinement of the McKay conjecture, proposed by Isaacs--Navarro in 2002, providing an important subcase of the celebrated McKay--Navarro conjecture with several local-global consequences.

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Problems on the conductor of finite group characters
math.RT 2025-12 conditional novelty 4.0 of 10

A survey of conductor problems for finite group characters that adds a new equivalence (combined Feit-deficiency conjecture iff Feit + cyclotomic deficiency) and two small p-rationality implications.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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O. Brunat and R. Nath , The Navarro conjecture for the alternating groups. Algebra Number Theory 15 (2021), no. 4, 821--862

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