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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   ${\mathbb C}[G^{u,v}]$   (  $u$  ,   $v\in W$  ) 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   $\Delta(k;\textbf{i})$   ([ 1 ,  8 ]). 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   $\{\Delta(k;\textbf{i})\}_{(m-2)r&lt;k\leq (m- 1)r}$   (  $m$   is given below) of the cluster algebra   $\mathbb C[L^{u,e}]$   [ 6 ,  8 ] 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 in Proposition 6.3 below. We also present the corresponding results for type   ${\rm A}_r$   though the results for all initial cluster variables have been obtained in [ 9 ].

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