Indecomposable summands of Foulkes modules

TL;DR

This paper investigates the modular structure of Foulkes modules for symmetric groups over fields of odd characteristic, identifying indecomposable summands, their vertices, and block structures, especially for low p-weight blocks.

Contribution

It characterizes the vertices of indecomposable summands of Foulkes modules and describes all summands in blocks of p-weight at most two, providing new structural insights.

Findings

Unique summand in the principal block when 2n < 3p

Description of summands in blocks of p-weight ≤ 2

Identification of vertices of indecomposable summands

Abstract

In this paper we study the modular structure of the permutation module $H^{(2^n)}$ of the symmetric group $S_{2n}$ acting on set partitions of a set of size $2n$ into $n$ sets each of size $2$, defined over a field of odd characteristic $p$. In particular we characterize the vertices of the indecomposable summands of $H^{(2^n)}$ and fully describe all of its indecomposable summands that lie in blocks of $p$-weight at most two. When $2n < 3p$ we show that there is a unique summand of $H^{(2^n)}$ in the principal block of $S_{2n}$ and that this summand exhibits many of the extensions between simple modules in its block.