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

Let F be a field and let E be the natural representation of SL₂(F). Given a vector space V, let Δ^(2,1N-1)V be the kernel of the multiplication map N V ⊗ V → N+1V. We construct an explicit SL₂(F)-isomorphism SymN-1E ⊗ N+1 Symᵈ⁺¹E Δ^(2,1N-1) Symᵈ E. This SL₂(F)-isomorphism is a modular lift of the q-binomial identity qN(N-1)/2[N]q d+1N+1q = s_(2,1N-1)(1,q,…, qᵈ), where s_(2,1N-1) is the Schur function for the partition (2,1N-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.