A new stability property of plethysm
Let \(G\) be a connected reductive algebraic group over \(\mathbb{C}\). Let \(\lambda\), \(\mu\), \(\nu\) be its dominant weights, \(\pi\), \(\theta\) partitions of an integer \(p\) and \(V_\nu\) a simple \(G\)-module with the highest weight \(\nu\). The author investigates the behaviour of the multiplicity of a simple \(G\)-module with the highest weight \(\mu+n\lambda\) in \(S_{\pi+n\theta}V_\nu\) when \(n\) tends to infinity. It is shown that this multiplicity always increases but not always becomes stable. A combinatorial criterion of stability is given. The principal result gives some necessary conditions for appearance of an irreducible representation in the plethysm.
scientific article; zbMATH DE number 949856