Cluster algebras of finite type via Coxeter elements and Demazure crystals of type B,C,D

Abstract

For a classical group G and a Coxeter element c of the Weyl group, it is known that the coordinate ring ℂ[Ge,c2] of the double Bruhat cell Ge,c2≔B∩B−c2B− has a structure of cluster algebra of finite type, where B and B− are opposite Borel subgroups. In this article, we consider the case G is of type Br, Cr or Dr and describe all the cluster variables in ℂ[Ge,c2] as monomial realizations of certain Demazure crystals.