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

In: Journal of Algebraic Combinatorics, 11.01.2020.

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• Accepted Manuscript

Accepted author manuscript, 528 KB, PDF document

### 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.
Original language English 52 Journal of Algebraic Combinatorics Accepted/In press - 11 Jan 2020

ID: 36489848