On permutation characters and Sylow $p$-subgroups of $\mathfrak{S}_n$

TL;DR

This paper investigates the structure of permutation modules induced by symmetric groups acting on cosets of Sylow p-subgroups, revealing their irreducible constituents and implications for related Hecke algebras.

Contribution

It determines the irreducible constituents of permutation modules for symmetric groups acting on Sylow p-subgroup cosets, a novel analysis in this context.

Findings

Identified the irreducible constituents of the permutation module

Calculated the number of irreducible representations of the associated Hecke algebra

Provided new insights into the representation theory of symmetric groups and Hecke algebras

Abstract

Let $p$ be an odd prime and let $n$ be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group $\mathfrak{S}_n$ on the cosets of a Sylow $p$-subgroup $P_n$. As a consequence, we determine the number of irreducible representations of the corresponding Hecke algebra $\mathcal{H}(\mathfrak{S}_n, P_n, 1_{P_n})$.