• DOI:10.1090/tran/8481
• Corpus ID: 53653670

Plethysms of symmetric functions and highest weight representations

@article{Boeck2021PlethysmsOS,
  title={Plethysms of symmetric functions and highest weight representations},
  author={Melanie de Boeck and Rowena Paget and Mark Wildon},
  journal={Transactions of the American Mathematical Society},
  year={2021}
}

• M. D. Boeck, R. Paget, M. Wildon
• Published 8 October 2018
• Mathematics
• Transactions of the American Mathematical Society

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.