Set families and Foulkes modules

TL;DR

This paper introduces new homomorphisms from Specht modules to Foulkes modules, providing a combinatorial characterization of minimal partitions labeling irreducible summands, with broader combinatorial implications.

Contribution

It constructs novel homomorphisms and offers a combinatorial description of minimal partitions in Foulkes modules, advancing understanding of their irreducible components.

Findings

New homomorphisms from Specht to Foulkes modules

Combinatorial description of minimal partitions

Combinatorial results on subset families

Abstract

We construct a new family of homomorphisms from Specht modules into Foulkes modules for the symmetric group. These homomorphisms 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 homomorphisms 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.