Cluster algebras of finite type via a Coxeter element and Demazure Crystals of type A

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.