Set families and Foulkes modules

Abstract. We construct a new family of homomorphisms from Specht modules into Foulkes modules for the symmetric group. These homo-morphisms are used to give a combinatorial description of the minimal partitions (in the dominance order) which label irreducible characters appearing as summands of the characters of Foulkes modules. The ho-momorphisms are defined using certain families of subsets of the natural numbers. These families are of independent interest; we prove a number of combinatorial results concerning them. 1