On the expansion formulas of cluster varieties from surfaces and their combinatorial properties
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917294788050944 |
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| author | Dinh, Vu Tung Lam Ip, Ivan Chi-Ho |
| author_facet | Dinh, Vu Tung Lam Ip, Ivan Chi-Ho |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21902 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the expansion formulas of cluster varieties from surfaces and their combinatorial properties Dinh, Vu Tung Lam Ip, Ivan Chi-Ho Combinatorics Representation Theory 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. |
| title | On the expansion formulas of cluster varieties from surfaces and their combinatorial properties |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2602.21902 |