Highest weight vectors in plethysms

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.

23 pages