Minimal numbers of linear constituents in Sylow restrictions for symmetric groups

TL;DR

This paper characterizes irreducible characters of symmetric groups with limited linear constituents in Sylow p-subgroup restrictions, providing explicit classifications and new Sylow branching coefficients for specific cases.

Contribution

It precisely identifies which irreducible characters have at most p linear constituents in their Sylow p-restriction, answering a previously open question and computing new coefficients.

Findings

Identified all irreducible characters with ≤ p linear constituents in Sylow p-restrictions.

Explicitly calculated Sylow branching coefficients for p=2 and almost hook partitions.

Provided a complete classification of such characters in symmetric groups.

Abstract

Let $p$ be any prime. We determine precisely those irreducible characters of symmetric groups which contain at most $p$ distinct linear constituents in their restriction to a Sylow $p$-subgroup, answering a question of Giannelli and Navarro. Moreover, we identify all of the linear constituents of such characters, and in the case $p = 2$ explicitly calculate a new class of Sylow branching coefficients for symmetric groups indexed by so-called almost hook partitions.