Plethysms of symmetric functions and representations of SL_2(C)

Let ∇λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL2(C). We make a systematic study of when there is an isomorphism ∇λSymℓE≅∇μSymmE of representations of SL2(C). Generalizing earlier results of King and Manivel, we classify all such isomorphisms when λ and μ are conjugate partitions and when one of λ or μ is a rectangle. We give a complete classification when λ and μ each have at most two rows or columns or is a hook partition and a partial classification when ℓ=m. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when ∇λSymℓE 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