3-Plethysms of homogeneous and elementary symmetric functions

TL;DR

This paper introduces a combinatorial tableau-based method to analyze coefficients in plethysms of Schur functions, providing new insights and extending results to broader cases using geometric and algebraic tools.

Contribution

It presents a novel plethystic tableau approach for understanding Schur function plethysms, extending results to general partitions via a Kronecker map.

Findings

Developed a combinatorial tableau framework for plethysm coefficients

Analyzed cases where $ u$ is a partition of 3 and $ extlambda$ has one part

Extended results to arbitrary partitions using a Kronecker map

Abstract

We introduce the new combinatorial approach of plethystic type of tableaux, as a method to understand coefficients of Schur functions appearing in plethysms $s_\nu[h_\lambda]$ and $s_{\nu}[e_{\lambda}]$, for any partitions $\lambda$ and $\nu$. We first give general results about this approach, then use results on tableaux, ribbon tableaux and integer points in polytopes to understand the case where $\nu$ is a partition of $3$ and $\lambda$ has one part. We then use a \textit{Kronecker map} to extend these results to any partition $\lambda$.