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arXiv · 2602.21902

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

Vu Tung Lam Dinh·Ivan Chi-Ho Ip
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Abstract

This paper explores the cluster algebra structure of the moduli space $\mathscr{A}_{\mathrm{SL}_{n+1},\mathbb{S}}$ of twisted $\mathrm{SL}_{n+1}$-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.

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Vu Tung Lam Dinh, Ivan Chi-Ho Ip. 2026-02-25. On the expansion formulas of cluster varieties from surfaces and their combinatorial properties. https://doi.org/10.48550/arxiv.2602.21902

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