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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 ${\mathbb C}[G^{u,v}]$ ($u$, $v\in W$) ofthe double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to the clusteralgebra ${\mathcal{A}}(\textbf{i})_{{\mathbb C}}$ and the initial clustervariables 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 clustervariables $\{\Delta(k;\textbf{i})\}_{(m-2)r<k\leq (m-1)r}$ ($m$ is somepositive integer) of the cluster algebra $\mathbb{C}[L^{u,e}]$ in terms ofmonomial realization of Demazure crystals, where $L^{u,e}$ is the reduceddouble 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 thecorresponding results for type ${\rm A}_r$ though the results for all initialcluster variables have been obtained by ourselves.

Cluster Variables on Double Bruhat Cells $G^{u,e}$ of Classical Groups and Monomial Realizations of Demazure Crystals