Combinatorial cluster expansion formulas from triangulated surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2018
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| _version_ | 1866917759310364672 |
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| author | Yurikusa, Toshiya |
| author_facet | Yurikusa, Toshiya |
| contents | We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of perfect matchings of angles. Our formula simplifies the cluster expansion formula given by Musiker-Schiffler-Williams in terms of perfect matchings of snake graphs. A key point of our proof is to give a bijection between perfect matchings of angles in some triangulated polygon and perfect matchings of the corresponding snake graph. Moreover, they also correspond bijectively with perfect matchings of the corresponding bipartite graph and minimal cuts of the corresponding quiver with potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_01567 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Combinatorial cluster expansion formulas from triangulated surfaces Yurikusa, Toshiya Combinatorics Commutative Algebra Representation Theory 13F60, 05C70, 05E15 We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of perfect matchings of angles. Our formula simplifies the cluster expansion formula given by Musiker-Schiffler-Williams in terms of perfect matchings of snake graphs. A key point of our proof is to give a bijection between perfect matchings of angles in some triangulated polygon and perfect matchings of the corresponding snake graph. Moreover, they also correspond bijectively with perfect matchings of the corresponding bipartite graph and minimal cuts of the corresponding quiver with potential. |
| title | Combinatorial cluster expansion formulas from triangulated surfaces |
| topic | Combinatorics Commutative Algebra Representation Theory 13F60, 05C70, 05E15 |
| url | https://arxiv.org/abs/1808.01567 |