Generalized Demazure Modules and Prime Representations in Type $D_n$

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 g. This family was introduced in [19,20] 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 g[t]; or equivalently a module for the maximal standard parabolic subalgebra in the affine Lie algebra g. In the case when g is of type An the problem was studied in [4], where it was shown that the graded limit is isomorphic to a level two Demazure module. In this paper we study the case when g is of type Dn. 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.

Quantum affine algebras, graded limits, and flags

“…A comparable study of HL-modules in other types is only partially explored. A first step was taken in [24] in type D n but it does not capture all the prime objects in the category F κ . There are important differences from the A n case and some new ideas seem to be necessary.…”

Quantum Affine Algebras, Graded Limits and Flags

“…A comparable study of HL-modules in other types is only partially explored. A first step was taken in 23 in type D n but it does not capture all the prime objects in the category F κ . There are…”

Prime representations in the Hernandez-Leclerc category: classical decompositions

“…After a suitable twist, which is referred to as the graded limit in the literature, many interesting families of graded representations of current algebras appear in this way. Among them are the Kirillov-Reshetikhin modules for current algebras (see [8,15,22]) and their fusion products [28] and also generalized Demazure modules appear in this context [7]. Inspired by the results of [1,2,28] the motivation of this paper is to fully understand the graded decompositions of the graded limits of the prime irreducible objects in the Hernandez-Leclerc category which we explain now in more detail.…”

“…In fact many families of Demazure modules and generalized Demazure modules appear as M ξ,λ for a suitable choice of ξ and λ; see for example [12,25] and [7].…”