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

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.