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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 ${\mathbb C}[G^{u,v}]$ ofthe double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to an uppercluster algebra $\bar{{\mathcal A}}({\bf i})_{{\mathbb C}}$ and the generalizedminors $\{\Delta(k;{\bf i})\}$ are the cluster variables belonging to a giveninitial 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 $G={\rmSL}_{r+1}({\mathbb C})$, $v=e$ and some special $u\in W$, we shall describe thegeneralized minors $\{\Delta(k;{\bf i})\}$ as summations of monomialrealizations of certain Demazure crystals.

Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A