arXiv2026
A. Bondal introduced the symplectic groupoid of triangular bilinear forms. This groupoid induces a Poisson structure on $\mathcal A_n$, the space of $n\times n$ unipotent upper-triangular matrices, governed by the classical $\mathfrak{so}(n)$ reflection equation. L. Chekhov and M. Shapiro described log-canonical coordinates on this symplectic groupoid via the $\mathcal A_n$-quiver. We introduce a birational Weyl-group action on the symplectic groupoid, generated by cluster transformations associated with cycles of the quiver. We prove that the matrix entries on $\mathcal A_n$ are invariant under this action. V. Fock and L. Chekhov defined a Poisson map $ϕ_n:\mathcal T_{g,s}\to\mathcal A_n$, where $s\in\{1,2\}$. Every $A\in\operatorname{Im}(ϕ_n)$ satisfies $\operatorname{rank}(A+A^T)\leq 4$, which provides a natural criterion for a cluster Poisson reduction of $\mathcal A_n$. The corresponding rank-condition locus has several irreducible components. We prove that the Weyl group acts transitively on these components and that the associated reductions are conjugate. Thus, it suffices to determine the reduction on a single component. For even $n$, we show that the longest Weyl-group element corresponds to a cluster Donaldson--Thomas transformation. The resulting theta basis is used to characterize the Weyl-invariant regular-function algebra in terms of the matrix entries on $\mathcal A_n$. In contrast, the $\mathcal A_n$-quiver admits no reddening sequence for odd $n$. Finally, we consider the $Σ_n$-quiver, a frozen extension of the $\mathcal A_{n+1}$-quiver used in J. Song's cluster realization of the $\imath$-quantum group of type $\mathrm{AI}_n$. For odd $n$, we identify the image of the classical limit of Song's embedding with a quotient of the corresponding Weyl-invariant Poisson algebra.