Sylow branching trees for symmetric groups

TL;DR

This paper introduces Sylow branching trees for symmetric groups, providing a combinatorial framework to describe Sylow branching coefficients for all irreducible characters of Sylow p-subgroups, extending prior work on linear characters.

Contribution

It develops a novel combinatorial tree-based method to analyze Sylow branching coefficients for symmetric groups, generalizing previous results on linear characters.

Findings

Describes Sylow branching coefficients using labelled p-ary trees.

Extends previous work from linear characters to all irreducible characters.

Provides a comprehensive combinatorial characterization for symmetric groups.

Abstract

Let $p\ge 5$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite symmetric group. To every irreducible character of $P$ we associate a collection of labelled, complete $p$-ary trees. The main results of this article describe Sylow branching coefficients for symmetric groups for all irreducible characters of $P$ in terms of some combinatorial properties of these trees, extending previous work on the linear characters of $P$.