A new order on integer partitions

TL;DR

This paper introduces a new relation on integer partitions linked to Schur positivity and plethysms, conjectures it as a partial order, and explores its properties and stability in the context of symmetric functions.

Contribution

It proposes a novel relation on partitions related to Schur positivity, conjectures it as a partial order, and analyzes its properties and stability.

Findings

Established properties of the new relation through explicit module inclusions

Conjectured the relation as a partial order and linked it to Foulkes conjecture

Proved stability properties for the number of irreducible modules as n increases

Abstract

Considering Schur positivity of differences of plethysms of homogeneous symmetric functions, we introduce a new relation on integer partitions. This relation is conjectured to be a partial order, with its restriction to one part partitions equivalent to the classical Foulkes conjecture. We establish some of the properties of this relation via the construction of explicit inclusion of modules whose characters correspond to the plethysms considered. We also prove some stability properties for the number of irreducible occurring in these modules as $n$ grows.