  Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$
and $B_-$ be two opposite Borel subgroups in $G$ and $W$ be the 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 $\bar{{\mathcal A}}({\bf i})_{{\mathbb C}}$ and the generalized
minors $\{\Delta(k;{\bf i})\}$ are the cluster variables belonging to a given
initial seed in ${\mathbb C}[G^{u,v}]$ [Berenstein A., Fomin S., Zelevinsky A.,
Duke Math. J. 126 (2005), 1-52, math.RT/0305434]. In the case $G={\rm
SL}_{r+1}({\mathbb C})$, $v=e$ and some special $u\in W$, we shall describe the
generalized minors $\{\Delta(k;{\bf i})\}$ as summations of monomial
realizations of certain Demazure crystals.
