Explicit forms of cluster variables on double Bruhat cells Gu,e of type B

Abstract

ABSTRACTLet G be a simply connected simple algebraic group over ℂ of type Br, B and B− be its two opposite Borel subgroups, and W be the associated Weyl group. For u, v∈W, it is known that the coordinate ring ℂ[Gu,v] of the double Bruhat cell is isomorphic to an upper cluster algebra and generalized minors Δ(k;i) are the cluster variables of ℂ[Gu,v][1]. It is also shown that ℂ[Gu,v] have a structure of cluster algebra [6]. In the case v = e, we shall describe the generalized minor Δ(k;i) explicitly.