On Truncated Weyl Modules

TL;DR

This paper investigates the structure of truncated Weyl modules, proving conjectures about their isomorphism to fusion products for certain weights, and explores their relations to other modules and algebraic structures.

Contribution

It proves the conjecture that truncated Weyl modules are isomorphic to fusion products for specific weights, and relates these modules to Chari-Venkatesh modules and Kirillov-Reshetikhin modules.

Findings

Proved the isomorphism for weights that are multiples of certain fundamental weights.

Established that truncated Weyl modules are quotients of fusion products of Kirillov-Reshetikhin modules.

Described the partitions corresponding to truncated Weyl modules explicitly.

Abstract

We study structural properties of truncated Weyl modules. A truncated Weyl module $W_N(\lambda)$ is a local Weyl module for $\mathfrak g[t]_N = \mathfrak g \otimes \frac{\mathbb C[t]}{t^N\mathbb C[t]}$, where $\mathfrak g$ is a finite-dimensional simple Lie algebra. It has been conjectured that, if $N$ is sufficiently small with respect to $\lambda$, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when $\lambda$ is a multiple of certain fundamental weights, including all minuscule ones for simply laced $\mathfrak g$. We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov-Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari-Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that $\mathfrak g=\mathfrak{sl}_2$ related to Demazure flags and chains of inclusions of truncated Weyl modules.