Highest Weight Vectors in Plethysms

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. In particular, we obtain alternative formulas for the multiplicities in these modules.