  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.
