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The McKay--Navarro conjecture for the prime 2

L. Ruhstorfer, A. A. Schaeffer Fry

TL;DR

The paper completes the McKay--Navarro conjecture for the prime $\ell=2$ by establishing the inductive McKay--Navarro conditions for the universal covering groups of the listed simple groups, including sporadic groups, alternating groups, and certain groups of Lie type. It develops and applies $\mathcal{H}$-equivariant bijections between the sets of $2'$-degree characters and their normalizers, leveraging Harish-Chandra theory, duality, and careful control of Galois actions on character values. The authors construct and analyze extension maps and stabilizers for principal-series labels, ensuring the required equivariance and extension properties across automorphisms and Galois actions. As a result, the Galois–McKay (McKay–Navarro) conjecture is validated for $\ell=2$ across the stated families, completing the inductive program initiated by Navarro–Späth–Vallejo and connecting to broader local-global character conjectures.

Abstract

We complete the proof of the McKay--Navarro conjecture (also known as the Galois--McKay conjecture) for the prime 2, by completing the proof of the inductive McKay--Navarro conditions introduced by Navarro--Späth--Vallejo in this situation.

The McKay--Navarro conjecture for the prime 2

TL;DR

The paper completes the McKay--Navarro conjecture for the prime by establishing the inductive McKay--Navarro conditions for the universal covering groups of the listed simple groups, including sporadic groups, alternating groups, and certain groups of Lie type. It develops and applies -equivariant bijections between the sets of -degree characters and their normalizers, leveraging Harish-Chandra theory, duality, and careful control of Galois actions on character values. The authors construct and analyze extension maps and stabilizers for principal-series labels, ensuring the required equivariance and extension properties across automorphisms and Galois actions. As a result, the Galois–McKay (McKay–Navarro) conjecture is validated for across the stated families, completing the inductive program initiated by Navarro–Späth–Vallejo and connecting to broader local-global character conjectures.

Abstract

We complete the proof of the McKay--Navarro conjecture (also known as the Galois--McKay conjecture) for the prime 2, by completing the proof of the inductive McKay--Navarro conditions introduced by Navarro--Späth--Vallejo in this situation.
Paper Structure (16 sections, 23 theorems, 15 equations)

This paper contains 16 sections, 23 theorems, 15 equations.

Table of Contents

  1. The Inductive McKay--Navarro Conditions
  2. Alternating Groups, Sporadic Groups, and Groups with Exceptional Schur Multipliers
  3. Notation and Preliminaries
  4. Characters
  5. Groups of Lie Type
  6. Harish-Chandra Theory
  7. Duality
  8. Automorphisms
  9. Odd-degree characters and rationality
  10. Recalling Previous Results
  11. The Characters $\eta^{(\sigma)}$ and $\gamma_{\lambda,\sigma}$
  12. Extension Maps and Galois Automorphisms
  13. More on Stabilizers and Extensions
  14. Equivariant Bijections for the Prime $2$
  15. An $\mathcal{H}$-equivariant map for $\widetilde{G}$
  16. ...and 1 more sections

Key Result

Theorem 1

Let $q$ be a power of an odd prime. Then the inductive McKay--Navarro conditions hold for the prime $\ell=2$ for the following simple groups:

Theorems & Definitions (43)

  • Theorem 1
  • Corollary 2
  • Definition 1.1: Definition 1.5 of NavarroSpathVallejo
  • Proposition 2.1
  • proof
  • Proposition 4.1: Malle--Späth, Malle MS16malle19
  • proof
  • Theorem 4.2: Schaeffer FrySFgaloisHC
  • Lemma 4.3
  • proof
  • ...and 33 more