Birational Weyl Group Action on the Symplectic Groupoid and Cluster Algebras

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 $φn$ 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 $n$.