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 Sym m Sym ℓ E ∼ = Sym ℓ Sym m E. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely Sym m Sym ℓ E ∼ = m Sym ℓ+m−1 E. 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, thenTo illustrate that the existence of such 'plethystic isomorphisms' is far from obvious, we end by proving that the generalisation ∇ λ Sym ℓ E ∼ = ∇ λ ′ Sym ℓ+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.