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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 anisomorphism $\nabla^\lambda \!\mathrm{Sym}^\ell \!E \cong \nabla^\mu\!\mathrm{Sym}^m \! E$ of representations of $\mathrm{SL}_2(\mathbb{C})$.Generalizing earlier results of King and Manivel, we classify all suchisomorphisms 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 hookpartition and a partial classification when $\ell = m$. As a corollary of amore general result on Schur functors labelled by skew partitions we alsodetermine all cases when $\nabla^\lambda \!\mathrm{Sym}^\ell \!E$ isirreducible. The methods used are from representation theory and combinatorics;in particular, we make explicit the close connection with MacMahon'senumeration of plane partitions, and prove a new $q$-binomial identity in thissetting.

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