On the decomposition of the Foulkes module
The Foulkes module $${H^{(a^b)}}$$ is the permutation module for the symmetric group Sab given by the action of Sab 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, 1r) with r ≥ 1 appears in $${H^{(a^b)}}$$ .