Birational Weyl Group Action on the Symplectic Groupoid and Cluster Algebras

TL;DR

This work develops a birational Weyl group action on the $\mathcal{A}_n$-quiver, establishing that the action preserves formal geodesic functions and that Weyl-invariant functions are generated by these geodesic functions. It then leverages this action to show transitivity of Hamiltonian reductions across irreducible components defined by rank conditions, enabling reductions from a single component. For even $n$, the longest Weyl element corresponds to a cluster DT-transformation, yielding a canonical (theta) basis, while odd $n$ admits no reddening sequence, signaling a structural distinction. The authors explicitly describe $O(\mathcal{X}_{|\mathcal{A}_4|})$ via canonical functions on a twice-punctured torus and articulate a broader canonical-basis conjecture for general $n$, supported by tropicalization and duality arguments. Overall, the paper links cluster-algebraic mutations, Poisson geometry, and Teichmüller-theoretic constructs to produce a concrete framework for invariants, reductions, and canonical bases in higher rank settings.

Abstract

A. Bondal introduced a symplectic groupoid of triangular bilinear forms. This induces the Poisson structure on $\mathcal{A}_n$, the space of $n \times n$ unipotent upper-triangular matrices. L. Chekhov and M. Shapiro described log-canonical coordinates on this symplectic groupoid via the $\mathcal{A}_n$-quiver. In this paper, we introduce a birational Weyl group action on the symplectic groupoid. It is generated by cluster transformations associated with certain cycles of the quiver. We prove that every matrix entry of $\mathcal{A}_n$ is invariant under this action. Conversely, we prove that matrix entries generate the Poisson subalgebra of Weyl group invariants in the ring of regular functions on the cluster variety $\mathcal X_{|\mathcal A_n|}$. V. Fock and L. Chekhov defined a Poisson map $φ_n$ from the Teichmüller space $\mathcal{T}_{g,s}$ with genus $g$ and $s \in \{1,2\}$ boundary components into $\mathcal{A}_n$. We aim to describe the cluster structure of $\text{Im}(φ_n)$ by applying a Hamiltonian reduction to $\mathcal A_n$. Since every element in $\text{Im}(φ_n)$ satisfies the rank condition $\text{rank}(A+A^T) \le 4$, it provides a natural criterion for our Hamiltonian reduction. The solution set of the rank condition has distinct irreducible components, so the reduction might be component-dependent. However, we show that the Weyl group acts transitively on these components. This implies that the Hamiltonian reductions are conjugate; thus, it suffices to determine the reduction on a single component. Furthermore, we prove that the longest element of the Weyl group corresponds to a cluster DT-transformation on the $\mathcal{A}_{2k}$-quiver. In contrast, we show that no reddening sequence exists for odd $n$.

Figures (27)

• Figure 1: 1 Examples of $\mathcal{A}_n$-quivers.
• Figure 2: Examples of doubled $\mathcal{A}_n$-quivers. Each color of vertices represents the main cycle.
• Figure 3: Geodesics in the upper half plane
• Figure 4: An example of an ideal triangulation on the reduced surface. The blue vertices are ideal points, green and red arcs are internal sides, and gray regions indicate the truncated infinite hyperboloids.
• Figure 5: The red line indicates a part of the path $P_{\gamma}$, and $[\gamma]$ is given by the expression $[\cdots L X_{z_i} R \cdots]$ from the fat graph. As $L$, $R$, and $X_{z_i}$ lie in $PSL(2,\mathbb{R})$, the matrix product is in $PSL(2,\mathbb{R})$.

Theorems & Definitions (134)

• Theorem A: Theorem \ref{['thm422']}
• Theorem B: Theorem \ref{['thm4318']}
• Theorem C: Corollary \ref{['cor523']}
• Theorem D: Theorems \ref{['thm631']} and \ref{['thm637']}
• Definition 2.1.1
• Example 2.1.2
• Definition 2.1.3
• Definition 2.1.4
• Theorem 2.1.5
• Corollary 2.1.6