Cluster realizations of Weyl groups and higher Teichmüller theory

Abstract

For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, we construct a family of weighted quivers $Q_m(\mathfrak{g})$ ($m\ge 2$) 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 $Q_{kh}(\mathfrak{g})$ ($k\ge 1$) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmüller 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 Teichmüller space of a general marked surface. We finally prove that this action coincides with the one constructed in [GS18] from the geometrical viewpoint.

70 pages, 28 figures. Final version. To appear in Selecta Mathematica, New Series

References in corpus (7)

• The quantum dilogarithm and representations quantum cluster varieties
• Cluster integrable systems, q-Painleve equations and their quantization
• Quantum geometry of moduli spaces of local systems and representation theory
• Synchronicity phenomenon in cluster patterns
• A conjecture on $C$-matrices of cluster algebras
• Generalized $q$-Painlevé VI systems of type $(A_{2n+1}+A_1+A_1{)}^{(1)}$ arising from cluster algebra
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Cited by in corpus (6)

• Cluster integrable systems and spin chains
• Generalized $q$-Painlevé VI systems of type $(A_{2n+1}+A_1+A_1{)}^{(1)}$ arising from cluster algebra
• Cluster realization of Weyl groups and $q$-characters of quantum affine algebras
• An affine Weyl group action on the basic hypergeometric series arising from the $q$-Garnier system
• Unbounded ${\mathfrak{sl}}_3$-laminations and their shear coordinates
• Wilson lines and their Laurent positivity