Highest Weight Vectors in Plethysms, II

Abstract

For an irreducible polynomial representation V of the general linear group \(\textrm{GL}_n(\mathbb {C})\), we realize its symmetric square \(S^2(V)\) and its alternating square \(\Lambda ^{\hspace{-1.5pt}{2}}(V)\) 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 \(\textrm{GL}_n(\mathbb {C})\) highest weight vectors which occur in \(V\otimes V\), \(S^2(V)\) and \(\Lambda ^{\hspace{-1.5pt}{2}}(V)\) respectively. In particular, we obtain new description of the multiplicities in \(S^2(V)\) and \(\Lambda ^{\hspace{-1.5pt}{2}}(V)\).