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

## TL;DR

This paper classifies when certain symmetric function representations of SL_2(C) are isomorphic, focusing on specific partition shapes, and explores irreducibility conditions using combinatorics and representation theory.

## Contribution

It provides a systematic classification of isomorphic symmetric function representations of SL_2(C) for specific partition types and establishes new combinatorial identities.

## Key findings

- Classified isomorphisms for conjugate partitions and rectangles.
- Determined all cases of irreducibility for these representations.
- Connected representation theory with MacMahon's plane partition enumeration.

## 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 $\nabla^\lambda \!\mathrm{Sym}^\ell \!E \cong \nabla^\mu \!\mathrm{Sym}^m \! E$ 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 $\nabla^\lambda \!\mathrm{Sym}^\ell \!E$ 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.