Plethysms of symmetric functions and representations of $\protect \mathrm{SL}_2({\protect \bf C})$

Paget, Rowena; Wildon, Mark

doi:10.5802/alco.150

Plethysms of symmetric functions and representations of

{SL}_{2} (C)

Paget, Rowena ¹; Wildon, Mark ²

¹ School of Mathematics, Statistics and Actuarial Science University of Kent Canterbury CT2 7FS, UK
² Department of Mathematics Royal Holloway University of London Egham TW20 0EX, UK

Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 27-68.

Abstract

Let $\nabla^{λ}$ denote the Schur functor labelled by the partition $λ$ and let $E$ be the natural representation of ${SL}_{2} (C)$ . We make a systematic study of when there is an isomorphism $\nabla^{λ} {Sym}^{ℓ} E ≅ \nabla^{μ} {Sym}^{m} E$ of representations of ${SL}_{2} (C)$ . 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 $\nabla^{λ} {Sym}^{ℓ} E$ 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.

Received: 2019-09-03
Revised: 2020-03-12
Accepted: 2020-09-10
Published online: 2021-02-16

DOI: 10.5802/alco.150

Classification: 05E05, 05E10, 20C30, 22E46, 22E47
Keywords: Plethysm, Hermite Reciprocity, Hook Content Formula

Author's affiliations:

Paget, Rowena ¹; Wildon, Mark ²

¹ School of Mathematics, Statistics and Actuarial Science University of Kent Canterbury CT2 7FS, UK
² Department of Mathematics Royal Holloway University of London Egham TW20 0EX, UK

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Paget, Rowena; Wildon, Mark. Plethysms of symmetric functions and representations of $\protect \mathrm{SL}_2({\protect \bf C})$. Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 27-68. doi : 10.5802/alco.150. https://alco.centre-mersenne.org/articles/10.5802/alco.150/

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