Field-independent Kronecker-plethysm isomorphisms

Abstract

We construct an explicit field-independent SL$_2$-equivariant isomorphism between an invariant space of tensors and a plethysm space. The existence of such an isomorphism was only known in characteristic 0, and only indirectly via character theory. Our isomorphism naturally extends the web of field-independent isomorphisms given by Hermite reciprocity, Hodge duality, and the Wronskian isomorphism. This is a characteristic free generalization of a classical situation in characteristic zero: certain rectangular Kronecker coefficients coincide with certain plethysm coefficients, and their non-negativiy proves the unimodality of the $q$-binomial coefficient. We also give a short combinatorial field-independent proof that the Hermite reciprocity map over the standard basis is a triangular matrix with 1s on the main diagonal.