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

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 $\mathcal{H}({\mathfrak{S}}_n,P_n,1_{P_n})$.

14 pages