Modular plethystic isomorphisms for two-dimensional linear groups

TL;DR

This paper constructs explicit plethystic isomorphisms for two-dimensional linear groups over arbitrary fields, generalizing classical results like Hermite reciprocity and the Wronskian isomorphism, and explores their limitations in prime characteristic fields.

Contribution

It provides new explicit plethystic isomorphisms for symmetric powers of the natural representation of SL_2, extending classical results to arbitrary fields and examining their limitations.

Findings

Explicit isomorphism between symmetric powers generalizing Hermite reciprocity.

Generalization of the Wronskian isomorphism to arbitrary fields.

Counterexamples showing certain plethystic isomorphisms do not hold in prime characteristic.

Abstract

Let $E$ be the natural representation of the special linear group $\mathrm{SL}_2(K)$ 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 $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \mathrm{Sym}_\ell \mathrm{Sym}^m E$. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \bigwedge^m \mathrm{Sym}^{\ell+m-1} E$. We also generalise a result first proved by King, by showing that if $\nabla^\lambda$ is the Schur functor for the partition $\lambda$ and $\lambda^\circ$ is the complement of $\lambda$ in a rectangle with $\ell+1$ rows, then $\nabla^\lambda \mathrm{Sym}^\ell E \cong \nabla^{\lambda^\circ} \mathrm{Sym}_\ell E$. To illustrate that the existence of such `plethystic isomorphisms' is far from obvious, we end by proving that the generalisation $\nabla^\lambda \mathrm{Sym}^\ell E \cong \nabla^{\lambda'} \mathrm{Sym}^{\ell + \ell(\lambda') - \ell(\lambda)}E$ 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.