A field-independent filtration of plethystic modules for ${\mathrm{SL}}_2(\mathbb{F})$ that categorifies a product rule for the Cartan subalgebra of ${\mathcal{U}}_q({\mathfrak{sl}}_2)$

Abstract

We lift a product rule in the Cartan subalgebra of quantum ${\mathfrak{sl}}_2$ to a filtration of the plethystic representation ${\Delta }^{(n,m)}{\mathrm{Sym}}^{d}E$ of the affine group scheme of the algebraic group ${\mathrm{SL}}_2$, where $E$ is the natural representation and ${\Delta }^{(n,m)}$ the Weyl functor. This is a significant step towards a categorification of quantum ${\mathfrak{sl}}_2$. Our filtration is an addition to a growing family of field-independent isomorphisms of ${\mathrm{SL}}_2$ representations that include Hermite reciprocity and the Wronskian isomorphism. It is the first such field-independent result requiring multiple filtration layers. It is proved by combinatorial techniques using the authors' symmetric functions model for Weyl modules.

23 pages, comments welcome