## Tokyo Journal of Mathematics

### Explicit Forms of Cluster Variables on Double Bruhat Cells $G^{u,e}$ of Type C

#### Abstract

Let $G=Sp_{2r}({\mathbb C})$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $C_r$, $B$ and $B_-$ its two opposite Borel subgroups, and $W$ 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 $G^{u,v}=BuB\cup B_-vB_-$ 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 ${\mathbb C}[G^{u,v}]${5}. In the case $v=e$, we shall describe the generalized minor $\Delta(k;\textbf{i})$ explicitly.

#### Article information

Source
Tokyo J. Math., Volume 39, Number 3 (2017), 643-678.

Dates
First available in Project Euclid: 6 October 2016

https://projecteuclid.org/euclid.tjm/1475723089

Digital Object Identifier
doi:10.3836/tjm/1475723089

Mathematical Reviews number (MathSciNet)
MR3634287

Zentralblatt MATH identifier
1378.13013

#### Citation

KANAKUBO, Yuki; NAKASHIMA, Toshiki. Explicit Forms of Cluster Variables on Double Bruhat Cells $G^{u,e}$ of Type C. Tokyo J. Math. 39 (2017), no. 3, 643--678. doi:10.3836/tjm/1475723089. https://projecteuclid.org/euclid.tjm/1475723089

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