Explicit Forms of Cluster Variables on Double Bruhat Cells G^{u,e} of type B

TL;DR

This paper explicitly describes the generalized minors, which are cluster variables, on double Bruhat cells of type B, enhancing understanding of their algebraic structure and cluster algebra relations.

Contribution

It provides explicit formulas for generalized minors on double Bruhat cells of type B, clarifying their cluster algebra structure.

Findings

Explicit formulas for generalized minors on G^{u,e}

Connection between minors and cluster variables

Enhanced understanding of cluster algebra structure in type B

Abstract

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $B_r$, $B$ and $B_-$ be its two opposite Borel subgroups, and $W$ be the associated Weyl group. For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to an upper cluster algebra $\overline{\mathcal{A}}(\textbf{i})_{{\mathbb C}}$ and the generalized minors $\Delta(k;\textbf{i})$ are the cluster variables of ${\mathbb C}[G^{u,v}]$[A.Berenstein, S.Fomin, A.Zelevinsky, Duke Math. J. 126 (2005), 1-52, arxiv:math.RT/0305434]. Recently, it is also shown that ${\mathbb C}[G^{u,v}]$ have structure of cluster algebra [K. R. Goodearl, M. T. Yakimov, arxiv:1602.00498 (2016)]. In the case $v=e$, we shall describe the generalized minor $\Delta(k;\textbf{i})$ explicitly.