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Plethysms of symmetric functions and highest weight representations

de Boeck, Melanie, Paget, Rowena E., Wildon, Mark (2021) Plethysms of symmetric functions and highest weight representations. Transactions of the American Mathematical Society, 374 . pp. 8013-8043. ISSN 0002-9947. (doi:10.1090/tran/8481) (KAR id:87807)


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.

Item Type: Article
DOI/Identification number: 10.1090/tran/8481
Subjects: Q Science > QA Mathematics (inc Computing science) > QA165 Combinatorics
Q Science > QA Mathematics (inc Computing science) > QA171 Representation theory
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Rowena Paget
Date Deposited: 28 Apr 2021 15:54 UTC
Last Modified: 21 Feb 2022 15:29 UTC
Resource URI: (The current URI for this page, for reference purposes)

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