Plethysms of symmetric functions and highest weight representations

Abstract:

Let $s_\nu \circ s_\mu$ denote the plethystic product of the Schur functions $s_\nu$ and $s_\mu$. In this article we define an explicit polynomial representation corresponding to $s_\nu \circ s_\mu$ 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 $\langle s_\nu \circ s_\mu , s_\lambda \rangle$ to be stable under insertion of new parts into $\mu$ and $\lambda$. We also characterize all maximal and minimal partitions $\lambda$ in the dominance order such that $s_\lambda$ appears in $s_\nu \circ s_\mu$ and determine the corresponding multiplicities using plethystic semistandard tableaux.