On the decomposition of the Foulkes module

TL;DR

This paper investigates the structure of the Foulkes module for symmetric groups, providing conditions for when simple modules appear or do not appear, with implications for Specht modules labeled by hook partitions.

Contribution

It offers a new sufficient condition for simple modules to have zero multiplicity in the Foulkes module, advancing understanding of its decomposition.

Findings

No Specht modules labeled by hook partitions appear in H^(a^b).

Provides a criterion for zero multiplicity of simple modules in the Foulkes module.

Enhances understanding of the module's decomposition structure.

Abstract

The Foulkes module H^(a^b) is the permutation module for the symmetric group S_ab given by the action of S_ab on the collection of set partitions of a set of size ab into b sets each of size a. The main result of this paper is a sufficient condition for a simple CS_{ab}-module to have zero multiplicity in H^(a^b). A special case of this result implies that no Specht module labelled by a hook partition (ab - r, 1^r) appears in H(a^b).