Geometric crystals and Cluster ensembles in Kac-Moody setting

Abstract: For a Kac-Moody group $G$, double Bruhat cells $G^{u,e}$ ($u$ is a Weyl group element) have positive geometric crystal structures. In arXiv:1210.2533, it is shown that there exist birational maps between `cluster tori' $\mathcal{X}{\Sigma}$ (resp. $\mathcal{A}{\Sigma}$) and $G_{\rm Ad}^{u,e}$ (resp. $G^{u,e}$), and they are extended to regular maps from cluster $\mathcal{X}$ (resp. $\mathcal{A}$) -varieties to $G_{\rm Ad}^{u,e}$ (resp. $G^{u,e}$). The aim of this article is to construct certain positive geometric crystal structures on the cluster tori $\mathcal{X}{\Sigma}$ and $\mathcal{A}{\Sigma}$ by presenting their explicit formulae. In particular, the geometric crystal structures on the tori $\mathcal{A}_{\Sigma}$ are obtained by applying the twist map. As a corollary, we see the sets of $\mathbb{Z}^T$-valued points of the cluster varieties have plural structures of crystals.