Cluster realizations of Weyl groups and higher Teichmüller theory

For a symmetrizable Kac–Moody Lie algebra g , we construct a family of weighted quivers Q m ( g ) ( m ≥ 2 ) whose cluster modular group Γ Q m ( g ) contains the Weyl group W ( g ) as a subgroup. We compute explicit formulae for the corresponding cluster A - and X -transformations. As a result, we obtain green sequences and the cluster Donaldson–Thomas transformation for Q m ( g ) in a systematic way when g is of finite type. Moreover if g is of classical finite type with the Coxeter number h , the quiver Q kh ( g ) ( k ≥ 1 ) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmüller space of a once-punctured disk with 2 k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmüller space of a general marked surface. We finally prove that this action coincides with the one constructed in Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint.