Resolutions and a Weyl Character formula for prime representations of quantum affine sl_{n+1}

TL;DR

This paper develops a Weyl character formula for prime irreducible representations of quantum affine sl_{n+1}, using BGG-type resolutions with local Weyl modules, connecting representation theory with cluster algebras.

Contribution

It introduces a Weyl character formula for prime quantum affine representations via BGG resolutions involving local Weyl modules, linking to cluster algebra variables.

Findings

Derived a closed Weyl character formula as an alternating sum of local Weyl module characters.

Established a connection between prime representations and cluster algebra variables.

Expressed prime level two Demazure module characters as combinations of level one Demazure modules.

Abstract

In this paper we study the family of prime irreducible representations of quantum affine $\lie{sl}_{n+1}$ which arise from the work of D. Hernandez and B. Leclerc. These representations can also be described as follows: the highest weight is a product of distinct fundamental weights with parameters determined by requiring that the representation be minimal by parts. We show that such representations admit a BGG-type resolution where the role of the Verma module is played by the local Weyl module. This leads to a closed formula (the Weyl character formula) for the character of the irreducible representation as an alternating sum of characters of local Weyl modules. In the language of cluster algebras our Weyl character formula describes an arbitrary cluster variable in terms of the generators $x_1,\cdots,x_n,x_1',\cdots, x_n'$ of an appropriate cluster algebra. Our results also exhibit the character of a prime level two Demazure module as an alternating linear combination of level one Demazure modules.