  Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$
and $B_-$ be its two opposite Borel subgroups. For two elements $u$, $v$ of the
Weyl group $W$, 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 a cluster
algebra $\mathcal{A}(\textbf{i})_{{\mathbb C}}$ [arXiv:math/0305434,
arXiv:1602.00498]. In the case $u=e$, $v=c^2$ ($c$ is a Coxeter element), the
algebra ${\mathbb C}[G^{e,c^2}]$ has only finitely many cluster variables. In
this article, for $G={\rm SL}_{r+1}(\mathbb{C})$, we obtain explicit forms of
all the cluster variables in $\mathbb{C}[G^{e,c^2}]$ by considering its
additive categorification via preprojective algebras, and describe them in
terms of monomial realizations of Demazure crystals.
