A new modular plethystic SL₂ (F) -isomorphism Sym^N-1E ^N+1 Sym^d+1E ^ (2, 1^{N-1) } Symᵈ E

Let F be a field and let E be the natural representation of SL₂ (F). Given a vector space V, let ^ (2, 1^{N-1) }V be the kernel of the multiplication map N V V ^N+1V. We construct an explicit SL₂ (F) -isomorphism Sym^N-1E ^N+1 Sym^d+1E ^ (2, 1^{N-1) } Symᵈ E. This SL₂ (F) -isomorphism is a modular lift of the q-binomial identity q^N (N-1) {2}Nq d+1N+1q = s (₂, ₁^₍-₁) (1, q, , qᵈ), where s (₂, ₁^₍-₁) is the Schur function for the partition (2, 1^N-1). This identity, which follows from our main theorem, implies the existence of an isomorphism when F is the field of complex numbers but it is notable, and not typical of the general case, that there is an explicit isomorphism defined in a uniform way for any field.