Modular plethystic isomorphisms for two-dimensional linear groups

Abstract

Let E be the natural representation of the special linear group SL2(K) over an arbitrary field K. We use the two dual constructions of the symmetric power when K has prime characteristic to construct an explicit isomorphism SymmSymℓE≅SymℓSymmE. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely SymmSymℓE≅⋀mSymℓ+m−1E. We also generalise a result first proved by King, by showing that if ∇λ is the Schur functor for the partition λ and λ∘ is the complement of λ in a rectangle with ℓ+1 rows, then ∇λSymℓE≅∇λ∘SymℓE. To illustrate that the existence of such ‘plethystic isomorphisms’ is far from obvious, we end by proving that the generalisation ∇λSymℓE≅∇λ′Symℓ+ℓ(λ′)−ℓ(λ)E of the Wronskian isomorphism, known to hold for a large class of partitions over the complex field, does not generalise to fields of prime characteristic, even after considering all possible dualities.