Inductive Galois–McKay Condition

• The inductive Galois–McKay condition is a reduction framework that refines the McKay conjecture by requiring Galois and automorphism equivariant bijections between ℓ'-characters and local subgroup representations.
• It integrates extension data via projective representations, ensuring compatibility between global character correspondences and the structure of local subgroups.
• The condition has been validated in key cases including prime 2 scenarios, defining characteristic groups of Lie type, and small-rank examples like PSL₂(q).

Searching arXiv for recent and foundational papers on the inductive Galois–McKay condition. The inductive Galois–McKay condition, also called the inductive McKay–Navarro condition, is the reduction-theoretic form of Navarro’s Galois refinement of the McKay conjecture. For a finite quasisimple group $G$, a prime $\ell$, a Sylow $\ell$-subgroup $Q$, and the distinguished Galois subgroup $H_\ell$ acting on $\ell'$-roots of unity by $\ell$-power maps, it requires a local subgroup $M$ containing $N_G(Q)$, an $\operatorname{Aut}(G)_Q\times H_\ell$-equivariant bijection between $\ell$0 and $\ell$1, and compatible extension data encoded by projective representations and character triples. In this form it strengthens the ordinary inductive McKay condition from automorphism-equivariant local-global character matching to a genuinely Galois-equivariant one (Ruhstorfer et al., 2021, Johansson, 2020).

1. Position within the McKay and Navarro conjectures

The classical McKay conjecture predicts that for a finite group $\ell$2 and a prime $\ell$3, the number of irreducible characters of degree prime to $\ell$4 in $\ell$5 equals the corresponding number for the normalizer of a Sylow $\ell$6-subgroup. In the notation used in the literature,

$\ell$7

and the conjecture compares $\ell$8 with $\ell$9, where $\ell$0 (Ruhstorfer et al., 2021).

Navarro’s refinement asks for more than equality of cardinalities. It introduces the subgroup

$\ell$1

consisting of those Galois automorphisms $\ell$2 such that, for some integer $\ell$3,

$\ell$4

for every root of unity $\ell$5 of order not divisible by $\ell$6, and predicts an $\ell$7-equivariant bijection between $\ell$8 and $\ell$9 (Ruhstorfer et al., 2022).

The inductive Galois–McKay condition is the simple-group reduction of this refinement. Recent work describes it as a system of conditions on the universal covering groups of non-abelian simple groups, analogous to the inductive McKay condition of Isaacs–Malle–Navarro, but with the Galois action built into both the character bijection and the extension-theoretic compatibility (Ruhstorfer et al., 2021).

2. Formal structure of the condition

A standard formulation starts with a finite quasisimple group $Q$0, a prime $Q$1, and a Sylow $Q$2-subgroup $Q$3. One asks for a proper $Q$4-stable subgroup $Q$5 with

$Q$6

and an $Q$7-equivariant bijection

$Q$8

such that corresponding characters lie over the same character of $Q$9 (Fry, 2020).

This is only the equivariant part. The full inductive Galois–McKay condition also imposes an extension condition. In one formulation, for each $H_\ell$0, one requires projective representations $H_\ell$1 and $H_\ell$2 of suitable semidirect products, defined over $H_\ell$3, whose factor sets take values in roots of unity, agree on the relevant intersection subgroup, and remain compatible after Galois twisting by $H_\ell$4 (Johansson, 2020). Other formulations express the same requirement in the language of character triples or $H_\ell$5-triples, using relations of the form

$H_\ell$6

where the relation $H_\ell$7 is encoded by associated projective representations with matching factor sets and matching scalar behavior on the relevant centralizer subgroup (Ruhstorfer et al., 20 Jun 2025).

A persistent source of confusion is the distinction between the equivariant bijection and the full inductive condition. Several papers separate these two layers explicitly. In particular, work on groups of Lie type proves that some previously known McKay bijections are already $H_\ell$8-equivariant and therefore are natural candidates for the inductive Galois–McKay condition, while emphasizing that the separate extension part is not automatic and may remain the main obstacle (Fry, 2020).

3. Harish–Chandra theoretic mechanisms and the equivariant bijection

For groups of Lie type, the decisive technical input is explicit control of Galois action on Harish–Chandra series. For a cuspidal pair $H_\ell$9, the irreducible characters in the Harish–Chandra series $\ell'$0 are parametrized by characters $\ell'$1 of a relative Weyl group, and the Galois action takes the form

$\ell'$2

with

$\ell'$3

The terms $\ell'$4, $\ell'$5, and the discrepancy between $\ell'$6 and $\ell'$7 are the obstructions to clean equivariance (Fry, 2020).

In the principal-series situation, the relevant local-global bijection is constructed by

$\ell'$8

where $\ell'$9 is a maximally split torus, $\ell$0, and $\ell$1 is an extension map for $\ell$2 (Fry, 2020). The central technical step is to prove, in the relevant cases, that the obstructions vanish or are controlled: $\ell$3 and $\ell$4 is controlled or trivial. This yields

$\ell$5

first on principal series and then on the full $\ell$6-character set in the cases treated there (Fry, 2020).

This mechanism shows why groups of Lie type are the principal testing ground for the inductive Galois–McKay condition. The problem is not merely to count $\ell$7-characters, but to synchronize automorphisms, Galois action, Harish–Chandra parametrizations, and extension maps in a single local-global construction.

4. Established cases: prime $\ell$8 and defining characteristic

A substantial part of the theory is now known in defining characteristic and at the prime $\ell$9.

Paper	Scope	Outcome
(Ruhstorfer et al., 2021)	$M$0, odd characteristic, untwisted groups without nontrivial graph automorphisms	Proves the inductive McKay–Navarro conditions for $M$1, $M$2, $M$3, $M$4, $M$5, and $M$6 under the stated hypotheses
(Johansson, 2020)	Defining characteristic for finite groups of Lie type	Completes the verification of the inductive McKay–Navarro condition for all finite groups of Lie type in defining characteristic
(Ruhstorfer et al., 2022)	Remaining prime-$M$7 simple groups	Completes the inductive McKay–Navarro conditions for $M$8 and deduces the McKay–Navarro conjecture for $M$9

For the prime $N_G(Q)$0, one major result established the inductive McKay–Navarro conditions for several families of finite simple groups of Lie type in odd characteristic, all untwisted and without nontrivial graph automorphisms. The proof used Harish–Chandra theory for disconnected groups, a detailed analysis of odd-degree characters, and construction of compatible extensions to inertia groups, with the main difficulty lying in the analogue of the extension condition from the Navarro–Späth–Vallejo framework (Ruhstorfer et al., 2021).

In defining characteristic, the remaining families were later handled by explicit control of the Galois action on Lusztig series and on local $N_G(Q)$1-characters. The resulting theorem verifies the inductive McKay–Navarro condition for groups with exceptional graph automorphisms, the Suzuki and Ree groups, $N_G(Q)$2 for $N_G(Q)$3, and groups with non-generic Schur multiplier, thereby completing the defining-characteristic case for all finite groups of Lie type (Johansson, 2020).

The prime-$N_G(Q)$4 program was then completed for the remaining simple groups, including sporadic groups, alternating groups, and the Lie types $N_G(Q)$5, $N_G(Q)$6, and $N_G(Q)$7 with graph automorphisms. This yields the global corollary that the McKay–Navarro conjecture holds for the prime $N_G(Q)$8 (Ruhstorfer et al., 2022).

5. Odd primes, type $N_G(Q)$9, and small-rank progress

For odd primes in cross-characteristic, the current picture is more partial. One line of work assembles tools for proving the inductive McKay–Navarro condition for groups of Lie type and odd primes $\operatorname{Aut}(G)_Q\times H_\ell$0, and proves the equivariance condition for $\operatorname{Aut}(G)_Q\times H_\ell$1 or $\operatorname{Aut}(G)_Q\times H_\ell$2. The framework uses Sylow $\operatorname{Aut}(G)_Q\times H_\ell$3-tori, $\operatorname{Aut}(G)_Q\times H_\ell$4-Harish–Chandra theory, generalized Gelfand–Graev representations, and equivariant extension maps. It yields an $\operatorname{Aut}(G)_Q\times H_\ell$5-equivariant bijection in type $\operatorname{Aut}(G)_Q\times H_\ell$6 and proves the full inductive condition on the unipotent-character part, while explicitly stating that the full conjecture is not finished in all cases (Ruhstorfer et al., 20 Jun 2025).

The same work formulates a general criterion reducing the inductive Galois–McKay problem to a combination of maximal extendibility, equivariant extension maps, stable transversals of characters, and a global-local bijection

$\operatorname{Aut}(G)_Q\times H_\ell$7

whose associated character triples satisfy a $\operatorname{Aut}(G)_Q\times H_\ell$8-relation. A further compatibility condition on the “Gallagher factor” upgrades this to the full Galois-twisted triple condition required by the inductive McKay–Navarro framework (Ruhstorfer et al., 20 Jun 2025).

At small rank, there is now a complete verification for $\operatorname{Aut}(G)_Q\times H_\ell$9. For a universal covering group $\ell$00 of $\ell$01, a prime $\ell$02, and $\ell$03 for $\ell$04, the condition is formulated via a $\ell$05-equivariant bijection

$\ell$06

for a suitable local subgroup $\ell$07, together with a central ordering of $\ell$08-triples. The resulting theorem proves that $\ell$09 satisfies the inductive Galois–McKay condition for every prime divisor $\ell$10 of $\ell$11 (Tapp, 29 Jul 2025).

6. Relation to ordinary inductive McKay and to nearby generalizations

The inductive Galois–McKay condition should be read against the background of the ordinary inductive McKay condition. A refinement of the ordinary theory shows that the inductive McKay condition yields more than numerical equality: it gives bijections on $\ell$12-characters compatible with automorphisms and preserving central isomorphism of character triples. In that setting one has relations of the form

$\ell$13

and the main theorem shows that this strengthened inductive condition globalizes from the universal covers of simple groups to arbitrary finite groups (Rossi, 2022). This does not yet incorporate Galois action, but it clarifies the structural strength expected of an inductive local-global correspondence.

Earlier precursor work proposed signed bijections between $\ell$14-degree characters of $\ell$15 and $\ell$16 compatible with restriction and induction modulo explicitly defined subgroups of virtual characters, and derived Isaacs–Navarro-type congruence consequences from those correspondences. That framework is not the modern inductive Galois–McKay condition, but it supplies a close character-theoretic precursor (Evseev, 2010).

More recent variants show how the local-global philosophy extends beyond the classical $\ell$17-character setting. For $\ell$18-solvable groups, one can replace $\ell$19-degree characters by $\ell$20-stable characters attached to a normal $\ell$21-series and obtain a McKay-type equality, together with a canonical bijection in the odd-order case; the paper explicitly describes this as a precursor or variant rather than a direct instance of the inductive Galois–McKay condition (Chang et al., 8 Dec 2025). In another direction, for a normal solvable subgroup $\ell$22 with Carter subgroup $\ell$23, there is a $\ell$24-equivariant bijection from Isaacs’ head characters $\ell$25 to $\ell$26, together with character-triple isomorphisms and degree control, giving a genuine analogue of the inductive McKay philosophy outside the Sylow-normalizer setting (Arranz, 13 Feb 2026).

Taken together, these developments indicate that the inductive Galois–McKay condition is best viewed not as an isolated conjectural statement, but as one node in a broader program of equivariant local-global correspondences. Its defining feature is that the local subgroup, the automorphism action, the Galois action, and the projective-representation data are all required to fit into a single compatible structure. That requirement is precisely what makes the condition strong enough to support reduction theorems and, in the cases already solved, to convert character-theoretic correspondences into proofs of global conjectures (Ruhstorfer et al., 2022).