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 $⟨s_{\nu }\circ s_{\mu },s_{\lambda }⟩$ 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.

34 pages