# Highest weight vectors in plethysms

## TL;DR

This paper provides explicit descriptions of highest weight vectors in certain plethysm modules of polynomial functions on matrices, focusing on the case k=3 for both symmetric and exterior powers.

## Contribution

It explicitly characterizes all highest weight vectors in the modules S^3(S^m(C^n)) and Λ^3(S^m(C^n)), advancing understanding of their structure.

## Key findings

- Explicit descriptions of highest weight vectors for k=3
- Realization of modules as polynomial functions on matrices
- Enhanced understanding of plethysm module structures

## Abstract

We realize the $\mathrm{GL}_n(\mathbb{C})$-modules $S^k(S^m(\mathbb{C}^n))$ and $\Lambda^k(S^m(\mathbb{C}^n))$ as spaces of polynomial functions on $n\times k$ matrices. In the case $k=3$, we describe explicitly all the $\mathrm{GL}_n(\mathbb{C})$-highest weight vectors which occur in $S^3(S^m(\mathbb{C}^n))$ and in $\Lambda^3(S^m(\mathbb{C}^n))$ respectively.