Plethysms of symmetric functions and highest weight representations

TL;DR

This paper introduces explicit polynomial representations for plethysm of symmetric functions, generalizes key results, and characterizes partitions with stable multiplicities using plethystic tableaux.

Contribution

It defines new representations linked to plethysm and extends known results, providing criteria for multiplicity stability and partition characterization.

Findings

Provided a sufficient condition for multiplicity stability under partition modifications.

Characterized all maximal and minimal partitions in the dominance order for plethysm.

Determined multiplicities using plethystic semistandard tableaux.

Abstract

Let $s_\nu \circ s_\mu$ denote the plethystic product of the Schur functions $s_\nu$ and $s_\mu$. In this article we define an explicit polynomial representation corresponding to $s_\nu \circ s_\mu$ with basis indexed by certain `plethystic' semistandard tableaux. Using these representations we prove generalizations of four results on plethysms due to Bruns--Conca--Varbaro, Brion, Ikenmeyer and the authors. In particular, we give a sufficient condition for the multiplicity $\langle s_\nu \circ s_\mu, s_\lambda\rangle$ to be stable under insertion of new parts into $\mu$ and $\lambda$. We also characterize all maximal and minimal partitions $\lambda$ in the dominance order such that $s_\lambda$ appears in $s_\nu \circ s_\mu$ and determine the corresponding multiplicities using plethystic semistandard tableaux.