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The Picky Conjecture for groups of Lie type

Gunter Malle, A. A. Schaeffer Fry

TL;DR

This work proves the Picky Conjecture for all quasi-simple groups of Lie type in non-defining characteristic $\ell \neq p$, via a local–global framework that uses Lusztig theory and a McKay-type bijection to compare $\operatorname{Irr}_{\ell'}(G)$ with $\operatorname{Irr}_{\ell'}(N)$ at picky $\ell$-elements. It additionally establishes a stronger version under natural Jordan decomposition assumptions and classifies semisimple picky elements, notably $2$- and $3$-elements, across Lie-type families. The paper extends these results to Suzuki and Ree groups, to $\ell \le 3$, and to exceptional covering groups, providing comprehensive coverage and explicit computations (where necessary) to verify the conjectures. Overall, the results reinforce the connections between the Picky Conjecture, McKay theory, and Galois-equivariant versions, clarifying how character values at picky elements behave under local-global correspondences in finite groups of Lie type.

Abstract

Recently, Moretó and Rizo proposed a conjecture, known as the Picky Conjecture, proposing new character correspondences extending the McKay Conjecture. We prove the Picky Conjecture for all quasi-simple groups of Lie type for non-defining primes. In favourable situations, we also obtain the stronger version postulating preservation of character values up to sign, and we show this stronger version holds in general when assuming certain natural properties of Lusztig's Jordan decomposition. Along the way, we complete the determination of semisimple picky elements in these groups by classifying picky 2- and 3-elements.

The Picky Conjecture for groups of Lie type

TL;DR

This work proves the Picky Conjecture for all quasi-simple groups of Lie type in non-defining characteristic , via a local–global framework that uses Lusztig theory and a McKay-type bijection to compare with at picky -elements. It additionally establishes a stronger version under natural Jordan decomposition assumptions and classifies semisimple picky elements, notably - and -elements, across Lie-type families. The paper extends these results to Suzuki and Ree groups, to , and to exceptional covering groups, providing comprehensive coverage and explicit computations (where necessary) to verify the conjectures. Overall, the results reinforce the connections between the Picky Conjecture, McKay theory, and Galois-equivariant versions, clarifying how character values at picky elements behave under local-global correspondences in finite groups of Lie type.

Abstract

Recently, Moretó and Rizo proposed a conjecture, known as the Picky Conjecture, proposing new character correspondences extending the McKay Conjecture. We prove the Picky Conjecture for all quasi-simple groups of Lie type for non-defining primes. In favourable situations, we also obtain the stronger version postulating preservation of character values up to sign, and we show this stronger version holds in general when assuming certain natural properties of Lusztig's Jordan decomposition. Along the way, we complete the determination of semisimple picky elements in these groups by classifying picky 2- and 3-elements.
Paper Structure (12 sections, 28 theorems, 23 equations, 3 tables)

This paper contains 12 sections, 28 theorems, 23 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The case of abelian Sylow subgroups
  3. The Setting
  4. Values on the local side
  5. Values on the global side
  6. On the Strong Picky Conjecture
  7. Suzuki and Ree groups
  8. Picky $2$-Elements
  9. The Picky Conjecture for $\ell\le3$ and for exceptional covering groups
  10. The case $\ell=2$
  11. The case $\ell=3$
  12. Exceptional covering groups

Key Result

Theorem 4

Let $G={\mathbf{G}}^F$ be a finite group of Lie type, where ${\mathbf{G}}$ is simple and simply connected in characteristic $p$ and $F$ is a Steinberg morphism. Then the Picky+ Conjecture holds for $G$ for any prime $\ell\geq 3$ different from $p$.

Theorems & Definitions (58)

  • Conjecture 1: Picky Conjecture
  • Definition 2: Picky+ Conjecture
  • Definition 3: Strong Picky Conjecture
  • Definition 3: Strong Picky Conjecture
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Lemma 2.1: Moretó--Rizo
  • Remark 2.2
  • Lemma 2.3: Moretó--Rizo
  • ...and 48 more