Highest weight vectors in plethysms (Q2200486)

Let \(V\) be a module over the general linear group \(\mathrm{GL}_n(\mathbb{C})\). Then, its symmetric powers \(S^k(V)\) and exterior powers \(\Lambda^k(V)\) are also \(\mathrm{GL}_n(\mathbb{C})\)-modules and we are interested in their decompositions into irreducible modules. This paper investigates the case where \(V=S^m(\mathbb{C}^n)\) and \(k=3\). By realizing the spaces \(S^3(V)\) and \(\Lambda^3(V)\) in concrete polynomial settings, the authors obtain explicit descriptions of highest weight vectors as well as multiplicity formulas for irreducible modules appearing in the decompositions of the spaces.