Plethysms of symmetric functions and highest weight representations

Abstract

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