Cluster algebras of finite type via Coxeter elements and Demazure crystals of type A

Abstract

Let G be a simply connected simple algebraic group over C, B and B− be its two opposite Borel subgroups. For two elements u, v of the Weyl group W, it is known that the coordinate ring C[Gu,v] of the double Bruhat cell Gu,v=BuB∩B−vB− is isomorphic to a cluster algebra A(i)C[2,12]. In the case u=e, v=c2 (c is a Coxeter element), the algebra C[Ge,c2] has only finitely many cluster variables. In this article, for G=SLr+1(C), we obtain explicit forms of all the cluster variables in C[Ge,c2] by considering its additive categorification via preprojective algebras, and describe them in terms of monomial realizations of Demazure crystals.