Highest Weight Vectors in Plethysms, II

For an irreducible polynomial representation V of the general linear group \(\textrm{GL}_n(\mathbb {C})\) ${\text{GL}}_n(\mathbb{C})$ , we realize its symmetric square \(S^2(V)\) $S^2(V)$ and its alternating square \(\Lambda ^{\hspace{-1.5pt}{2}}(V)\) ${\mathrm{\Lambda }}^2(V)$ as spaces of polynomial functions. In the case when V is labeled by a Young diagram with at most 2 rows, we describe explicitly all the \(\textrm{GL}_n(\mathbb {C})\) ${\text{GL}}_n(\mathbb{C})$ highest weight vectors which occur in \(V\otimes V\) $V\otimes V$ , \(S^2(V)\) $S^2(V)$ and \(\Lambda ^{\hspace{-1.5pt}{2}}(V)\) ${\mathrm{\Lambda }}^2(V)$ respectively. In particular, we obtain new description of the multiplicities in \(S^2(V)\) $S^2(V)$ and \(\Lambda ^{\hspace{-1.5pt}{2}}(V)\) ${\mathrm{\Lambda }}^2(V)$ .