Geometric crystals and Cluster ensembles in Kac-Moody setting

TL;DR

This paper constructs explicit positive geometric crystal structures on cluster tori associated with Kac-Moody groups, extending the interplay between geometric crystals and cluster varieties in the Kac-Moody setting.

Contribution

It explicitly formulates positive geometric crystal structures on cluster tori and demonstrates their relation via the twist map, expanding the geometric crystal framework in Kac-Moody groups.

Findings

Explicit formulae for geometric crystal structures on cluster tori

Extension of crystal structures via the twist map

Multiple crystal structures on integer points of cluster varieties

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.