Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions

TL;DR

This paper develops a combinatorial rule to identify the extremal partitions in plethysms of Schur functions, solving a key open problem and proving conjectures related to lexicographically extreme constituents.

Contribution

It introduces a new combinatorial method to determine all maximal and minimal partitions in plethysms of Schur functions, and proves related conjectures.

Findings

Established a combinatorial rule for extremal partitions in plethysms.

Proved three conjectures of Agaoka on lexicographically greatest and least partitions.

Showed the multiplicity of the least constituent can be arbitrarily large.

Abstract

This paper proves a combinatorial rule giving all maximal and minimal partitions $\lambda$ such that the Schur function $s_\lambda$ appears in a plethysm of two arbitrary Schur functions. Determining the decomposition of these plethysms has been identified by Stanley as a key open problem in algebraic combinatorics. As corollaries we prove three conjectures of Agaoka on the partitions labelling the lexicographically greatest and least Schur functions appearing in an arbitrary plethysm. We also show that the multiplicity of the Schur function labelled by the lexicographically least constituent may be arbitrarily large. The proof is carried out in the symmetric group and gives an explicit non-zero homomorphism corresponding to each maximal or minimal partition.