**Plethysms of symmetric functions and highest weight representations.** / de Boeck, Melanie; Paget, Rowena; Wildon, Mark.

Research output: Contribution to journal › Article › peer-review

E-pub ahead of print

**Plethysms of symmetric functions and highest weight representations.** / de Boeck, Melanie; Paget, Rowena; Wildon, Mark.

Research output: Contribution to journal › Article › peer-review

de Boeck, M, Paget, R & Wildon, M 2021, 'Plethysms of symmetric functions and highest weight representations', *Transactions of the American Mathematical Society*. https://doi.org/10.1090/tran/8481

de Boeck, M., Paget, R., & Wildon, M. (2021). Plethysms of symmetric functions and highest weight representations. *Transactions of the American Mathematical Society*. https://doi.org/10.1090/tran/8481

de Boeck M, Paget R, Wildon M. Plethysms of symmetric functions and highest weight representations. Transactions of the American Mathematical Society. 2021 Aug 23. https://doi.org/10.1090/tran/8481

@article{41d094f6b8d0459a8758b537a54cf3a0,

title = "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.",

author = "{de Boeck}, Melanie and Rowena Paget and Mark Wildon",

year = "2021",

month = aug,

day = "23",

doi = "10.1090/tran/8481",

language = "English",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

}

TY - JOUR

T1 - Plethysms of symmetric functions and highest weight representations

AU - de Boeck, Melanie

AU - Paget, Rowena

AU - Wildon, Mark

PY - 2021/8/23

Y1 - 2021/8/23

N2 - 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.

AB - 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.

U2 - 10.1090/tran/8481

DO - 10.1090/tran/8481

M3 - Article

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

ER -