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Plethysms of symmetric functions and representations of SL_2(C)

Paget, Rowena E., Wildon, Mark (2021) Plethysms of symmetric functions and representations of SL_2(C). Algebraic Combinatorics, 4 (1). pp. 27-68. ISSN 2589-5486. (doi:10.5802/alco.150) (KAR id:87806)

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Official URL
https://doi.org/10.5802/alco.150

Abstract

Let ∇λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL2(C). We make a systematic study of when there is an isomorphism ∇λSymℓE≅∇μSymmE of representations of SL2(C). Generalizing earlier results of King and Manivel, we classify all such isomorphisms when λ and μ are conjugate partitions and when one of λ or μ is a rectangle. We give a complete classification when λ and μ each have at most two rows or columns or is a hook partition and a partial classification when ℓ=m. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when ∇λSymℓE is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon’s enumeration of plane partitions, and prove a new q-binomial identity in this setting

Item Type: Article
DOI/Identification number: 10.5802/alco.150
Uncontrolled keywords: Plethysm, Hermite Reciprocity, Hook Content Formula
Subjects: Q Science > QA Mathematics (inc Computing science)
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 15:46 UTC
Last Modified: 30 Apr 2021 09:06 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/87806 (The current URI for this page, for reference purposes)
Paget, Rowena E.: https://orcid.org/0000-0001-8088-4421
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