Explicit forms of cluster variables on double Bruhat cells G^u,e of type B

ABSTRACT

Let G be a simply connected simple algebraic group over ℂ of type B_r, 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[G^u,v] of the double Bruhat cell $G^{u,v}=BuB\cup B_{-}v{B}_{-}$ is isomorphic to an upper cluster algebra $\overline{\mathcal{𝒜}}{\left(i\right)}_{ℂ}$ and generalized minors Δ(k;i) are the cluster variables of ℂ[G^u,v][1]. It is also shown that ℂ[G^u,v] have a structure of cluster algebra [6]. In the case v = e, we shall describe the generalized minor Δ(k;i) explicitly.

KEYWORDS:

• Cluster algebras
• double Bruhat cells
• generalized minors

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F60
• 20G42

Acknowledgment

I would like to express my sincere gratitude to T. Nakashima for his helpful comments and wide-ranging discussions.