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 $G^{u,v}=BuB\cap B_{-}v{B}_{-}$ is isomorphic to the cluster algebra $\mathcal{A}(\mathbf{\text{i}}{)}_{\mathbb{C}}$ and the initial cluster variables of $\mathbb{C}[G^{u,v}]$ are the generalized minors $\mathrm{\Delta }(k;\mathbf{\text{i}})$ by Berenstein, Fomin, Zelevinsky, Goodearl and Yakimov. In the case that a classical group $G$ is of type ${\mathrm{B}}_r$, ${\mathrm{C}}_r$ or ${\mathrm{D}}_r$, we shall describe the non-trivial last $r$ initial cluster variables $\{\mathrm{\Delta }(k;\mathbf{\text{i}}){\}}_{(m-2)r<k\le (m-1)r}$ ($m$ 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 $\mathrm{\Delta }(k;\mathbf{\text{i}})$ on $G^{u,e}$ and on $L^{u,e}$ is described as well. We also present the corresponding results for type ${\mathrm{A}}_r$ though the results for all initial cluster variables have been obtained by ourselves.