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Minimal numbers of linear constituents in Sylow restrictions for symmetric groups

Bim Gustavsson, Stacey Law

TL;DR

This work determines which irreducible characters of the symmetric group $S_n$ restrict to a Sylow $p$-subgroup $P$ with at most $p$ distinct linear constituents, resolving a question of Giannelli–Navarro. It develops a two-pronged analysis: for $p=2$ it yields a complete, explicit classification tied to almost-hook partitions and derives a precise formula for Sylow branching coefficients in that case, including a bijection between hook-like partitions and linear constituents. For odd primes $p$, the authors provide a sharp, case-based criterion depending on the $p$-adic structure of $n$ and the partition class, together with explicit computations of linear constituents via plethysm and wreath-product machinery. The results illuminate the positivity and distribution of Sylow branching coefficients, relate them to the Littlewood–Richardson and plethysm frameworks, and have implications for understanding $\Phi(S_n,p)$ and related questions for $A_n$. Overall, the paper advances the explicit understanding of how irreducible characters of $S_n$ decompose when restricted to Sylow subgroups and showcases how combinatorics of partitions governs these Sylow branching phenomena.

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.

Minimal numbers of linear constituents in Sylow restrictions for symmetric groups

TL;DR

This work determines which irreducible characters of the symmetric group restrict to a Sylow -subgroup with at most distinct linear constituents, resolving a question of Giannelli–Navarro. It develops a two-pronged analysis: for it yields a complete, explicit classification tied to almost-hook partitions and derives a precise formula for Sylow branching coefficients in that case, including a bijection between hook-like partitions and linear constituents. For odd primes , the authors provide a sharp, case-based criterion depending on the -adic structure of and the partition class, together with explicit computations of linear constituents via plethysm and wreath-product machinery. The results illuminate the positivity and distribution of Sylow branching coefficients, relate them to the Littlewood–Richardson and plethysm frameworks, and have implications for understanding and related questions for . Overall, the paper advances the explicit understanding of how irreducible characters of decompose when restricted to Sylow subgroups and showcases how combinatorics of partitions governs these Sylow branching phenomena.

Abstract

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

Paper Structure

This paper contains 8 sections, 19 theorems, 37 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The representation theory of the symmetric group
  4. Wreath products
  5. Plethysm coefficients
  6. Sylow branching coefficients for almost hooks
  7. Proof of \ref{['thm:main-p=2']}
  8. Odd primes

Key Result

Theorem 1.1

Let $n\in\mathbb{N}$ and $\lambda\in\mathcal{P}(n)$. Let $P\in\operatorname{Syl}_2(S_n)$. Then $|\operatorname{Lin}(\chi^\lambda\downarrow_P)|=2$ if and only if one of the following holds: Moreover, in each of the above cases, $\left\langle\chi^\lambda\downarrow_P,\psi\right\rangle=1$ for all $\psi\in\operatorname{Lin}(\chi^\lambda\downarrow_P)$.

Figures (2)

  • Figure 1: Some subgroups of $S_{2^k}$.
  • Figure 2: Using the Littlewood--Richardson rule to verify $c^\lambda_{\alpha,\alpha}>0$ where $\lambda=\mu\sqcup\nu\sqcup(1^{2^k-|\mu|-|\nu|})$, $\mu=(2^{k-1})$, $\nu=(4)$ and $\alpha=\mathsf{ah}_{2^{k-1}}(2^{k-2}-3)$: the shape $\lambda$ is split into shaded and unshaded parts. The shaded part illustrates $\alpha$, and the numbers filled into the unshaded part have reverse row word of content $\alpha$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3: Mac95
  • Theorem 2.4
  • Lemma 2.5
  • ...and 26 more