Modular isomorphisms of $\mathrm{SL}_2(\mathbb{F})$-plethysms for Weyl modules labelled by hook partitions

Abstract

Let $\Delta^\lambda$ be the Weyl functor for the partition $\lambda$ and let $E$ be the natural $2$-dimensional representation of $\mathrm{SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. We give an explicit isomorphism showing that any $\mathrm{SL}_2(\mathbb{F})$-plethysm $\Delta^{(M,1^N)}\mathrm{Sym}^d E$ factors as a tensor product of two simpler $\mathrm{SL}_2(\mathbb{F})$-plethysms, each defined using only symmetric powers. This result categorifies Stanley's Hook Content Formula for hook-shaped partitions and proves a conjecture of Mart\'inez--Wildon (2024). In a similar spirit we categorify the classical binomial identity $\binom{a}{b}\binom{b}{c}=\binom{a}{c}\binom{a-c}{b-c}$, obtaining a new family of $\mathrm{SL}_2(\mathbb{F})$-isomorphisms between tensor products of plethysms. Our methods are characteristic independent and provide a framework that is broadly applicable to the study of isomorphisms between plethystic representations of $\mathrm{SL}_2(\mathbb{F})$.