Local Weyl modules and skew Howe duality

Abstract

The skew $(\mathfrak{gl}_{n}, \mathfrak{gl}_{r})$ Howe duality states that the exterior algebra $\Lambda(\mathbb{C}^{nr})$ admits a multiplicity-free decomposition under the natural actions of $\mathfrak{gl}_{n}\times \mathfrak{gl}_{r}$. In this paper, by using certain Lagrange interpolation polynomials of degree $r-1$, we extend the action of $\mathfrak{gl}_{n}$ on $\Lambda(\C^{nr})$ to its loop algebra $L(\mathfrak{gl}_{n})$. View $\Lambda(\C^{nr})$ as a module for the loop algebra $L(\mathfrak{sl}_{n})$ of $\mathfrak{sl}_{n}$ by taking restriction. We prove that every highest weight vector of $\mathfrak{gl}_{n}\times \mathfrak{gl}_{r}$ in $\Lambda(\C^{nr})$ generates a local Weyl module of $L(\mathfrak{sl}_{n})$. Furthermore, we obtain in this way an explicit realization of all local Weyl modules for $L(\mathfrak{sl}_{n})$.