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 Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint.

Appendix A. Description of functions on \(\mathop {\mathrm {Conf}}\nolimits _3{\mathcal {A}}_G\)

In this appendix, we describe the composite morphism

$$\begin{aligned} {\widetilde{\beta }}:H\times H\times U^-_{*}\overset{\beta }{\rightarrow } \mathop {\mathrm {Conf}}\nolimits _3^{*}{{\mathcal {A}} }_G\hookrightarrow \mathop {\mathrm {Conf}}\nolimits _3{{\mathcal {A}} }_G \end{aligned}$$

in terms of the regular functions on these spaces. We can use this observation to derive the description of cluster ${{\mathcal {A}} }$-coordinates on the configuration space $\mathop {\mathrm {Conf}}\nolimits _3 {{\mathcal {A}} }_G$ (in particular, the data $\mu _{s, i}$) in Theorem 5.14 from Le’s paper [34] (See Remark A.4).

Let ${\mathbb {C}}[G]$ be the coordinate algebra of the semisimple simply-connected algebraic group G over ${\mathbb {C}}$. Then ${\mathbb {C}}[G]$ is considered as a $G\times G$-module by

$$\begin{aligned} ((g_1, g_2).F)(g):=F\big (g_1^Tgg_2\big ) \end{aligned}$$

for $g, g_1, g_2\in G$, $F\in {\mathbb {C}}[G]$. Note that, for $f\in V^{*}$, $u\in V$ and $g_1, g_2\in G$,

$$\begin{aligned} (g_1, g_2). c_{f, u}^V=c_{g_1.f, g_2.u}^V. \end{aligned}$$

Definition A.1

For $\lambda \in P_+$, set

$$\begin{aligned} V^-(\lambda ):=\big \{c_{f,v_{w_0\lambda }}^{\lambda }\mid f\in V(\lambda )^{*}\big \}. \end{aligned}$$

Then $V(\lambda )\rightarrow V^-(\lambda ), u\mapsto c_{u^{\vee },v_{w_0\lambda }}^{\lambda }$ gives an isomorphism of $G(\simeq G\times 1)$-modules. Set

$$\begin{aligned} {\mathbb {C}}[{{\mathcal {A}} }_G]:={\mathbb {C}}[G]^{1\times U^-}=\bigoplus _{\lambda \in P_{+}}V^-(\lambda ). \end{aligned}$$

Then ${\mathbb {C}}[{{\mathcal {A}} }_G]$ is a ${\mathbb {C}}$-subalgebra of ${\mathbb {C}}[G]$ and the elements of ${\mathbb {C}}[{{\mathcal {A}} }_G]$ determine well-defined functions on ${{\mathcal {A}} }_G$. Hence the elements of

$$\begin{aligned} ({\mathbb {C}}[G]^{1\times U_-})^{\otimes 3}=\bigoplus _{\lambda , \mu , \nu \in P_{+}}V^-(\lambda )\otimes V^-(\mu )\otimes V^-(\nu ) \end{aligned}$$

give well-defined functions on ${\mathcal {A}}_G\times {\mathcal {A}}_G\times {\mathcal {A}}_G$. Set

$$\begin{aligned} {\mathbb {C}}[\mathop {\mathrm {Conf}}\nolimits _3{\mathcal {A}}_G]:=\bigoplus _{\lambda , \mu , \nu \in P_{+}}(V^-(\lambda )\otimes V^-(\mu )\otimes V^-(\nu ))^{\Delta G}, \end{aligned}$$

here $\Delta G$ is the diagonal subgroup of $G\times G\times G$, which is isomorphic to G. Then ${\mathbb {C}}[\mathop {\mathrm {Conf}}\nolimits _3{\mathcal {A}}_G]$ is a ${\mathbb {C}}$-subalgebra of $({\mathbb {C}}[G]^{1\times U^-})^{\otimes 3}$ and the elements of ${\mathbb {C}}[\mathop {\mathrm {Conf}}\nolimits _3{\mathcal {A}}_G]$ determine well-defined functions on $\mathop {\mathrm {Conf}}\nolimits _3{\mathcal {A}}_G$.

Now the morphism ${\widetilde{\beta }} :H\times H\times U^-_{*}\rightarrow \mathop {\mathrm {Conf}}\nolimits _3{\mathcal {A}}_G$ induces a ${\mathbb {C}}$-algebra homomorphism

$$\begin{aligned} {\widetilde{\beta }}^{*}:{\mathbb {C}}[\mathop {\mathrm {Conf}}\nolimits _3({\mathcal {A}}_G)]\rightarrow {\mathbb {C}}[H\times H\times U^-_{*}] \end{aligned}$$

which is given by pull-back. We consider a description of the images of the elements of $(V^-(\lambda )\otimes V^-(\mu )\otimes V^-(\nu ))^{\Delta G}$ under ${\widetilde{\beta }}^{*}$:

Theorem A.2

Let $\lambda , \mu , \nu \in P_{+}$ and $F\in (V^-(\lambda )\otimes V^-(\mu )\otimes V^-(\nu ))^{\Delta G}$. Then

$$\begin{aligned} ({\widetilde{\beta }}^{*}(F))(h_1, h_2, u_-)=h_1^{\mu }s_G^{\mu }h_2^{\nu }s_G^{\nu }\langle f,u_-.v_{\nu }\rangle \end{aligned}$$

(A.1)

for $(h_1, h_2, u_-)\in H\times H\times U_-^{*}$, here $f\in V(\nu )^{*}$ is uniquely determined by the expression

$$\begin{aligned} F=\Delta _{w_0\lambda , w_0\lambda }\otimes \Delta _{\mu , w_0\mu }\otimes c^{\nu }_{f,v_{w_0\nu }} + \sum _{i, j, k}c^{\lambda }_{f_{i},v_{w_0\lambda }}\otimes c^{\mu }_{f_{j},v_{w_0\mu }}\otimes c^{\nu }_{f_k,v_{w_0\nu }}, \end{aligned}$$

(A.2)

where $f_i$, $f_j$ in the second term of the right-hand side are weight vectors of $V(\lambda )^{*}$, $V(\mu )^{*}$, respectively, such that $f_i\otimes f_j\not \in V(\lambda )_{w_0\lambda }^{*}\otimes V(\mu )_{\mu }^{*}$. In particular,

$$\begin{aligned} ({\widetilde{\beta }}^{*}(F))(h_1, h_2, u_-)=ah_1^{\mu }s_G^{\mu }h_2^{\nu }s_G^{\nu }\Delta _{-w_0\lambda -\mu , \nu }(u_-) \end{aligned}$$

if $-w_0\lambda -\mu \in W({\mathfrak {g}})\cdot \nu$, here $f=af_{-w_0\lambda -\mu }\in V(\nu )^{*}$ with $a\in {\mathbb {C}}$.

Proof

We can regard F as a function on ${\mathcal {A}}_G\times {\mathcal {A}}_G\times {\mathcal {A}}_G$, denoted by ${\widehat{F}}$, because $(V^-(\lambda )\otimes V^-(\mu )\otimes V^-(\nu ))^{\Delta G}$ is a subspace of $V^-(\lambda )\otimes V^-(\mu )\otimes V^-(\nu )$. Let ${\widehat{\beta }}$ be a morphism

$$\begin{aligned} H\times H\times U_-^{*}\rightarrow {\mathcal {A}}_G\times {\mathcal {A}}_G\times {\mathcal {A}}_G, (h_1, h_2, u_-)\mapsto (U^-, h_1\overline{w_0}U^-, u_-h_2\overline{w_0} U^-). \end{aligned}$$

Then we have

$$\begin{aligned} {\widehat{\beta }}^{*}({\widehat{F}})={\widetilde{\beta }}^{*}(F). \end{aligned}$$

We calculate the pull-back of the first term of (A.2) by ${\widehat{\beta }}$ and that of the second term separately. First,

$$\begin{aligned}&\left( {\widehat{\beta }}^{*}\left( \Delta _{w_0\lambda , w_0\lambda }\otimes \Delta _{\mu , w_0\mu }\otimes c^{\nu }_{f,v_{w_0\nu }}\right) \right) (h_1, h_2, u_-)\\&\quad =\left( \Delta _{w_0\lambda , w_0\lambda }\otimes \Delta _{\mu , w_0\mu }\otimes c^{\nu }_{f,v_{w_0\nu }}\right) (U^-, h_1\overline{w_0} U^-, u_-h_2\overline{w_0} U^-)\\&\quad =\langle f_{w_0\lambda }, v_{w_0\lambda }\rangle \langle f_{\mu }, h_1\overline{w_0}. v_{w_0\mu }\rangle \langle f, u_-h_2\overline{w_0}. v_{w_0\nu }\rangle \\&\quad =s_G^{\mu }s_G^{\nu }\langle f_{\mu }, h_1. v_{\mu }\rangle \langle f, u_-h_2. v_{\nu }\rangle \\&\quad =h_1^{\mu }s_G^{\mu }h_2^{\nu }s_G^{\nu }\langle f,u_-.v_{\nu }\rangle . \end{aligned}$$

Next, we have

$$\begin{aligned}&\left( {\widehat{\beta }}^{*}\left( \sum _{i, j, k}c^{\lambda }_{f_{i},v_{w_0\lambda }}\otimes c^{\mu }_{f_{j},v_{w_0\mu }}\otimes c^{\nu }_{f_k,v_{w_0\nu }}\right) \right) (h_1, h_2, u_-)\\&\quad =\left( \sum _{i, j, k}c^{\lambda }_{f_{i},v_{w_0\lambda }}\otimes c^{\mu }_{f_{j},v_{w_0\mu }}\otimes c^{\nu }_{f_k,v_{w_0\nu }}\right) (U^-, h_1\overline{w_0} U^-, u_-h_2\overline{w_0} U^-)\\&\quad =\sum _{i, j, k}\langle f_{i},v_{w_0\lambda }\rangle \langle f_{j},h_1\overline{w_0}. v_{w_0\mu }\rangle \langle f_k, u_- h_2\overline{w_0}.v_{w_0\nu }\rangle \\&\quad =\sum _{i, j, k}h_1^{\mu }s_G^{\mu }\langle f_{i},v_{w_0\lambda }\rangle \langle f_{j},v_{\mu }\rangle \langle f_k, u_- h_2\overline{w_0}.v_{w_0\nu }\rangle \\&\quad =0, \end{aligned}$$

here the last equality follows from our assumption on $f_{i}$ and $f_{j}$. These calculations complete the proof of (A.1). The second statement in the theorem follows from (A.1) and the fact that $f\in V(\nu )_{-w_0\lambda -\mu }^{*}$. $\square$

Remark A.3

If one of $\lambda , \mu$ or $\nu$ is equal to 0, then $\tau _1+w_0\tau _2=0$, where $\{\tau _1, \tau _2\}=\{\lambda , \mu ,\nu \}{\setminus }\{0\}$ as far as $(V^-(\lambda )\otimes V^-(\mu )\otimes V^-(\nu ))^{\Delta G}\ne 0$. Note that, in this case, the condition $-w_0\lambda -\mu \in W({\mathfrak {g}})\cdot \nu$ is satisfied.

Remark A.4

The case-by-case checking shows that Le’s cluster ${{\mathcal {A}} }$-coordinates on the configuration space $\mathop {\mathrm {Conf}}\nolimits _3 {{\mathcal {A}} }_G$ are chosen from the space of the form $(V^-(\lambda )\otimes V^-(\mu )\otimes V^-(\nu ))^{\Delta G}$ whose weight datum $(\lambda , \mu , \nu )$ satisfies the condition $-w_0\lambda -\mu \in W({\mathfrak {g}})\cdot \nu$. Hence, by Theorem A.2, the images of the cluster ${{\mathcal {A}} }$-coordinates on $\mathop {\mathrm {Conf}}\nolimits _3 {{\mathcal {A}} }_G$ under ${\widetilde{\beta }}^{*}$ are determined up to constant multiple only from the corresponding weight data $(\lambda , \mu , \nu )$.

Therefore, to obtain the description in Theorem 5.14, we only have to know the weight data $(\lambda , \mu , \nu )$ for the cluster ${{\mathcal {A}} }$-coordinates on $\mathop {\mathrm {Conf}}\nolimits _3 {{\mathcal {A}} }_G$ and their values at ${\widetilde{\beta }}(1, 1, u_-)$ for some $u_-\in U^{*}_-$. The former can be read from [34, p.33 (1)–(4), p.85–86 (1)–(5), p.119–120 (1)–(4), p.121 (1)–(5), p.121–122 (1)–(5)] (see also [34, Observations 3.5 and 5.3]), and the latter can be derived from [34, Propositions 3.1, 4.7 and 5.7].

Remark A.5

In fact, the set of elements of $V(\nu )^{*}$ which can appear as f in (A.2) is exactly equal to

$$\begin{aligned} \left\{ f\in V(\nu )_{-w_0\lambda -\mu }^{*}\Bigg |\begin{array}{l}f_s^{(k)}.f=0 \ \text { for all }k>\langle \alpha _s^{\vee }, -w_0\lambda \rangle , s\in S, \text { and}\\ e_s^{(k)}.f=0 \ \text { for all }k>\langle \alpha _s^{\vee }, \mu \rangle , s\in S\end{array}\right\} . \end{aligned}$$

We omit the proof of this fact since we do not use it in this paper. See, for example, [37, Proposition 31.2.6].