Plethysms of symmetric functions and highest weight representations

Let sν ◦ sµ denote the plethystic product of Schur functions and sµ. In this article we define an explicit polynomial representation corresponding to with basis indexed by certain ‘plethystic’ semistandard tableaux. Using these representations prove generalizations four results on plethysms due Bruns–Conca–Varbaro, Brion, Ikenmeyer authors. particular, give a sufficient condition for multiplicity hsν sµ, sλi be stable under insertion new parts into µ λ. We also characterize all maximal minimal partitions λ in dominance order such that sλ appears ◦sµ determine multiplicities using