Cluster Variables on Double Bruhat Cells G^(u,e) of Classical Groups and Monomial Realizations of Demazure Crystals

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ its two opposite Borel subgroups, and $W$ the associated Weyl group. It is shown that the coordinate ring ${\mathbb C}[G^{u,v}]$ ($u$, $v\in W$) of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to the cluster algebra ${\mathcal{A}}(\textbf{i})_{{\mathbb C}}$ and the initial cluster variables of ${\mathbb C}[G^{u,v}]$ are the generalized minors $\Delta(k;\textbf{i})$ by Berenstein, Fomin, Zelevinsky, Goodearl and Yakimov. In the case that a classical group $G$ is of type ${\rm B}_r$, ${\rm C}_r$ or ${\rm D}_r$, we shall describe the non-trivial last $r$ initial cluster variables $\{\Delta(k;\textbf{i})\}_{(m-2)r<k\leq (m-1)r}$ ($m$ is some positive integer) of the cluster algebra $\mathbb{C}[L^{u,e}]$ in terms of monomial realization of Demazure crystals, where $L^{u,e}$ is the reduced double Bruhat cell of type $(u,e)$. The relation between $\Delta(k;\textbf{i})$ on $G^{u,e}$ and on $L^{u,e}$ is described as well. We also present the corresponding results for type ${\rm A}_r$ though the results for all initial cluster variables have been obtained by ourselves.