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.
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