Demazure Modules, Chari-Venkatesh Modules and Fusion Products

TL;DR

This paper proves the parameter independence of certain fusion products of Demazure modules and Chari-Venkatesh modules, providing explicit relations and descriptions for these modules in the context of simple Lie algebras.

Contribution

It establishes the independence of fusion products of Demazure modules from parameters and characterizes the fusion of Chari-Venkatesh modules as a Chari-Venkatesh module, with explicit relations.

Findings

Fusion product of Demazure modules is parameter-independent.

Fusion of Chari-Venkatesh modules yields a Chari-Venkatesh module.

Provides explicit defining relations and descriptions for these modules.

Abstract

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $\theta$. Given two non-negative integers $m$, $n$, we prove that the fusion product of $m$ copies of the level one Demazure module $D(1,\theta)$ with $n$ copies of the adjoint representation ${\rm ev}_0 V(\theta)$ is independent of the parameters and we give explicit defining relations. As a consequence, for $\mathfrak{g}$ simply laced, we show that the fusion product of a special family of Chari-Venkatesh modules is again a Chari-Venkatesh module. We also get a description of the truncated Weyl module associated to a multiple of $\theta$.