Cluster realizations of Weyl groups and higher Teichmüller theory

• Rei Inoue ^[1] ; Tsukasa Ishibashi ^[2] ; Hironori Oya ^[3]
  1. [1] Chiba University

     Chiba University

     Chūō-ku, Japón

     
  2. [2] Kyoto University

     Kyoto University

     Kamigyō-ku, Japón

     
  3. [3] Shibaura Institute of Technology

     Shibaura Institute of Technology

     Japón

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 3, 2021
• Idioma: inglés
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• Resumen

  • For a symmetrizable Kac–Moody Lie algebra g, we construct a family of weighted quivers Qm(g) (m≥2) whose cluster modular group ΓQm(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 Qm(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 Qkh(g) (k≥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 Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint.