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.