Plethysms of symmetric functions and representations of SL_2(C). / Wildon, Mark; Paget, Rowena.

In: Journal of Algebraic Combinatorics, 11.01.2020.

Research output: Contribution to journalArticlepeer-review

Forthcoming

### Standard

In: Journal of Algebraic Combinatorics, 11.01.2020.

Research output: Contribution to journalArticlepeer-review

### Harvard

Wildon, M & Paget, R 2020, 'Plethysms of symmetric functions and representations of SL_2(C)', Journal of Algebraic Combinatorics.

### APA

Wildon, M., & Paget, R. (Accepted/In press). Plethysms of symmetric functions and representations of SL_2(C). Journal of Algebraic Combinatorics.

### Vancouver

Wildon M, Paget R. Plethysms of symmetric functions and representations of SL_2(C). Journal of Algebraic Combinatorics. 2020 Jan 11.

### Author

Wildon, Mark ; Paget, Rowena. / Plethysms of symmetric functions and representations of SL_2(C). In: Journal of Algebraic Combinatorics. 2020.

### BibTeX

title = "Plethysms of symmetric functions and representations of SL_2(C)",
abstract = "Let $\nabla^\lambda$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^\lambda \Sym^\ell E \cong \nabla^\mu \Sym^m E$ of representations of $\mathrm{SL}_2(\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $\lambda$ and $\mu$ are conjugate partitions and when one of $\lambda$ or $\mu$ is a rectangle. We give a complete classification when $\lambda$ and $\mu$ each have at most two rows or columns or is a hook partition and a partial classification when $\ell = m$. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when $\nabla^\lambda \Sym^\ell E$ 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.",
author = "Mark Wildon and Rowena Paget",
year = "2020",
month = jan,
day = "11",
language = "English",
journal = "Journal of Algebraic Combinatorics",
issn = "0925-9899",
publisher = "Springer Netherlands",

}

### RIS

TY - JOUR

T1 - Plethysms of symmetric functions and representations of SL_2(C)

AU - Wildon, Mark

AU - Paget, Rowena

PY - 2020/1/11

Y1 - 2020/1/11

N2 - Let $\nabla^\lambda$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^\lambda \Sym^\ell E \cong \nabla^\mu \Sym^m E$ of representations of $\mathrm{SL}_2(\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $\lambda$ and $\mu$ are conjugate partitions and when one of $\lambda$ or $\mu$ is a rectangle. We give a complete classification when $\lambda$ and $\mu$ each have at most two rows or columns or is a hook partition and a partial classification when $\ell = m$. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when $\nabla^\lambda \Sym^\ell E$ 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.

AB - Let $\nabla^\lambda$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^\lambda \Sym^\ell E \cong \nabla^\mu \Sym^m E$ of representations of $\mathrm{SL}_2(\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $\lambda$ and $\mu$ are conjugate partitions and when one of $\lambda$ or $\mu$ is a rectangle. We give a complete classification when $\lambda$ and $\mu$ each have at most two rows or columns or is a hook partition and a partial classification when $\ell = m$. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when $\nabla^\lambda \Sym^\ell E$ 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.

M3 - Article

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

ER -