On the expansion formulas of cluster varieties from surfaces and their combinatorial properties

This paper explores the cluster algebra structure of the moduli space ${\mathcal{A}}_{{\mathrm{S}\mathrm{L}}_{n+1},\mathbb{S}}$ of twisted ${\mathrm{S}\mathrm{L}}_{n+1}$-local systems on a surface. We derive general recurrence relations for cluster variables arising from flips of a triangulation, corresponding to specific sequences of mutations. Our approach is grounded in a detailed combinatorial analysis over the standard $n$-triangulated $m$-gon (with explicit calculations for $n=1,2$). As a generalization, the non-simply-laced $G_2$ type is also considered. We prove the “well-triangulated” property for cluster mutations under flips, which provides a combinatorial framework for understanding the stability and transformation rules of these cluster algebra structures, and compute the monomial counts for the cluster expansion formula.