Generalized Demazure Modules and Prime Representations in Type $D_n$

TL;DR

This paper investigates the graded limits of prime representations of quantum affine algebras of type Dn, showing they are often generalized Demazure modules and providing their presentations and characters.

Contribution

It demonstrates that certain graded limits of prime representations are generalized Demazure modules and offers explicit presentations and character formulas for these modules.

Findings

Limit modules are often generalized Demazure modules.

Presented explicit module presentations.

Computed graded characters in terms of level two Demazure modules.

Abstract

The goal of this paper is to understand the graded limit of a family of irreducible prime representations of the quantum affine algebra associated to a simply-laced simple Lie algebra $\mathfrak{g}$. This family was introduced by David Hernandez and Bernard Leclerc in the context of monoidal categorification of cluster algebras. The graded limit of a member of this family is an indecomposable graded module for the current algebra $\mathfrak{g}[t]$; or equivalently a module for the maximal standard parabolic subalgebra in the affine Lie algebra $\widehat{\mathfrak{g}}$. In this paper we study the case when $\mathfrak{g}$ is of type $D_n$. We show that in certain cases the limit is a generalized Demazure module, i.e., it is a submodule of a tensor product of level one Demazure modules. We give a presentation of these modules and compute their graded character (and hence also the character of the prime representations) in terms of Demazure modules of level two.