Sylow branching trees for symmetric groups

TL;DR

The paper addresses how irreducible characters of Sylow $p$-subgroups $P$ of symmetric groups restrict to $P$ by introducing a combinatorial encoding via labelled $p$-ary trees. It establishes that for $p\ge5$ the Sylow branching sets $\Omega(\theta)$ for $\theta\in\mathrm{Irr}(P_n)$ are governed by simple box-structures $\mathcal{B}_n(t)$ and a punctured variant, with exact descriptions when the associated tree statistic $\mathcal{V}(\theta)=1$ and precise bounds otherwise; the key quantities $\gamma_0(\theta)$ and $\gamma_1(\theta)$ derived from the trees drive the results. The work develops a robust framework using a tree-to-character correspondence, the $\star$-product of partition sets, and detailed LR-rule analysis to derive exact values of $M(\theta)=n-\gamma_0(\theta)$ and $m(\theta)$ (including puncture conditions), and proves that asymptotically almost all partitions occur as constituents in restrictions, i.e. $|\Omega(n)|/|\mathcal{P}(n)|\to1$. These results extend prior linear-character analyses and provide deep structural insight into Brauer-type restrictions for symmetric groups, with clear distinctions from the $p=2,3$ cases.

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 $P$.

Figures (5)

• Figure 1: Examples illustrating Definition \ref{['def:trees-stats']} with $p=3$. Here, and in subsequent figures, filled circles $\bullet$ indicate (unnamed) vertices in $\mathsf{H}(T)$, i.e. those with label in $[1,p-1]$, and unfilled circles $\circ$ indicate (unnamed) vertices not in $\mathsf{H}(T)$, i.e. those with label in $\{0,p\}$. The label of each vertex is written above it. This is to aid with visualising and counting which vertices are 'people' (Definition \ref{['def:trees-stats']}(a)) in our figures illustrating such labelled trees.
• Figure 2: Subgroups of $S_{p^k}$, where $Q:=P_{p^{k-1}}$
• Figure 3: Examples of $\theta\in\operatorname{Irr}(P_{p^3})$ where $p=5$, together with a representative admissible tree $T\in\mathcal{T}(\theta)$, tree statistics from Definition \ref{['def:trees-stats']} and information on their Sylow branching coefficients.
• Figure 4: Examples of admissible trees and condition (b) from Theorem \ref{['thm:punctures']}. Recall that a vertex $x$ is a person if and only if $\mathsf{v}(x)=1$, i.e. the label of $x$ belong to $[1,p-1]$. In the first subfigure we have omitted all vertex labels.
• Figure 5: Those $\theta\in\operatorname{Irr}(P_{p^3})$ for which $\Omega(\theta)$ is punctured, drawn using 3-ary trees. Here, $\varepsilon\in[1,p-1]$. We write $\mathcal{X}(a;b;c)$ for $\mathcal{X}( \mathcal{X}(\phi_a;\phi_b); \phi_c)$ and $\mathbbm{1}=\mathbbm{1}_{p^2}$.

Theorems & Definitions (101)

• Lemma 2.1
• proof
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Example 2.6
• Proposition 2.7
• Proposition 2.8
• Proposition 2.9