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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.21902 |
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Table of Contents:
- 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.