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

Abstract

Let $\hat{I}{”}^{\hat{I}}»$ be the Weyl functor for the partition $\hat{I}»$ 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 $\hat{I}{”}^{(M,1^N)}{\mathrm{Sym}}^{d}E$ 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 Martínez--Wildon (2024). In a similar spirit we categorify the classical binomial identity $\left(\genfrac{}{}{0ex}{}{a}{b}\right)\left(\genfrac{}{}{0ex}{}{b}{c}\right)=\left(\genfrac{}{}{0ex}{}{a}{c}\right)\left(\genfrac{}{}{0ex}{}{a-c}{b-c}\right)$, 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 ${\mathrm{SL}}_2(\mathbb{F})$.

19 pages. Minor revisions after comments kindly given by Darij Grinberg