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 $B_-$ 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 ${\mathbb C}[G^{u,v}]$[Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52, math.RT/0305434]. In the case $v=e$, we shall describe the generalized minor $\Delta(k;\textbf{i})$ explicitly.