Cluster realizations of Weyl groups and higher Teichm\"uller theory

TL;DR

This paper constructs quivers related to symmetrizable Kac-Moody algebras whose cluster groups contain Weyl groups, and connects these structures to higher Teichmüller spaces, providing explicit formulas and demonstrating their geometric actions.

Contribution

It introduces a systematic construction of quivers for Kac-Moody algebras, linking cluster theory with higher Teichmüller spaces and Weyl group actions, with explicit formulas and geometric interpretations.

Findings

Explicit formulas for cluster transformations of constructed quivers.

Systematic construction of green sequences and Donaldson-Thomas transformations.

Identification of quivers with cluster structures of higher Teichmüller spaces.

Abstract

For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, we construct a family of weighted quivers $Q_m(\mathfrak{g})$ ($m \geq 2$) whose cluster modular group $\Gamma_{Q_m(\mathfrak{g})}$ contains the Weyl group $W(\mathfrak{g})$ as a subgroup. We compute explicit formulae for the corresponding cluster $\mathcal{A}$- and $\mathcal{X}$-transformations. As a result, we obtain green sequences and the cluster Donaldson-Thomas transformation for $Q_m(\mathfrak{g})$ in a systematic way when $\mathfrak{g}$ is of finite type. Moreover if $\mathfrak{g}$ is of classical finite type with the Coxeter number $h$, the quiver $Q_{kh}(\mathfrak{g})$ ($k \geq 1$) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichm\"uller space of a once-punctured disk with $2k$ marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichm\"uller space of a general marked surface. We finally prove that this action coincides with the one constructed in [GS18] from the geometrical viewpoint.