arXiv:1704.02520v1 [math.QA] 8 Apr 2017 RESOLUTIONS AND A WEYL CHARACTER FORMULA FOR PRIME REPRESENTATIONS OF QUANTUM AFFINE sl n+1 MATHEUS BRITO AND VYJAYANTHI CHARI Abstract. In this paper we study the family of prime irreducible representations of quantum affine sln+1 which arise from the work of [26], [28]. 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 [26] our Weyl character formula describes an arbitrary cluster variable in terms of the generators x1, ··· ,xn,x ′ 1 , ··· ,x ′ n of an appropriate cluster algebra. Together with [3] our results also exhibit the character of a prime level two Demazure module as an alternating linear combination of level one Demazure modules. Introduction The study of the category  F q of finite–dimensional representations of a quantum affine algebra goes back nearly thirty years and continues to be of significant interest. In recent years, there has been new insight into the subject coming from its connections with cluster algebras through the work of [26], [28], [36] and also from KLR algebras through the work of [30, 31]. In [3] the connection with the theory of Demazure modules occurring in an affine Lie algebras was expanded beyond level one; the connection with level one Demazure modules was first made in [14] using the theory of Weyl modules introduced in [13]. From the point of view of representation theory, a natural problem is to understand the character of a finite–dimensional representation. In this context the theory of q–characters introduced in [21] and developed extensively in [5, 20, 22, 23, 24, 25, 33, 34, 35] has been a powerful tool. It has led to many important developments in the area, such as, the theory of T – systems and the connections with cluster algebras. The q–character is known combinatorially for some important families of representations such as the Kirillov–Reshetikhin modules and more generally the minimal affinizations. The q–characters are hard to compute and closed formulae are not known for a general simple object of  F q . The q–character is known to be multiplicative and hence as a first step it would be interesting to determine closed formulae for the prime irreducible representations in  F q ; namely an irreducible representation which cannot be written as a non–trivial tensor product of objects of  F q . In fact, the formulae known so far are for precisely such modules. Unfortunately however no classification of prime objects is known and even finding new families of examples is not easy. The input from cluster MB was partially supported by CNPq, grant 205281/2014-1. VC was partially supported by DMS-1303052. 1