The Picky and Subnormalizer Conjectures for symmetric groups

TL;DR

This work extends McKay-type local-global conjectures to symmetric groups by focusing on picky elements—$p$-elements that lie in a unique Sylow $p$-subgroup—and constructing precise bijections between appropriate irreducible character sets on groups and their normalizers or subnormalizers. The authors develop a detailed framework based on the $p$-core tower and core-quotient theory, combined with hook-length/Murnaghan–Nakayama techniques and a careful analysis of Sylow normalizers, to prove a strong Picky Conjecture for $\mathsf{S}_n$ (Theorem A) and a strong form of the Subnormalizer Conjecture for $p=2$ (Theorem B). They classify picky elements in $\mathsf{S}_n$ and show how to construct explicit bijections that preserve $p$-parts and control character values on picky elements, thereby generalizing local-global principles in a non-Lie-type family. The results rely on a blend of combinatorial partition theory, block-structure arguments, and inertia-group/Gallagher-type reasoning, and they pave the way for extensions to odd primes and broader group families.

Abstract

A new type of conjectures on characters of finite groups, related to the McKay conjecture, have recently been proposed. In this paper, we study these conjectures for symmetric groups.

Theorems & Definitions (70)

• Theorem A
• Theorem B
• Theorem 2.1: Murnaghan--Nakayama rule
• Theorem 2.2
• proof
• Proposition 2.3
• proof
• Corollary 2.4
• Proposition 2.5
• proof