On the decomposition of the Foulkes module

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 ${\mathbb{C} S_{ab}}$ -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) with r ≥ 1 appears in ${H^{(a^b)}}$ .