Construction of an explicit isomorphism in characteristic-free settings demonstrates new relationships, highlighting connections between isomorphisms and coefficients.
We construct an explicit field-independent SL₂-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.
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Ikenmeyer et al. (2025) studied this question.
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