Minimal Matchings for dP3 Cluster Variables
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910595985440768 |
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| author | Chiang, Judy Hsin-Hui Musiker, Gregg Nguyen, Son |
| author_facet | Chiang, Judy Hsin-Hui Musiker, Gregg Nguyen, Son |
| contents | In previous work [LM17], Tri Lai and the second author studied a family of subgraphs of the dP3 brane tiling, called Aztec castles, whose dimer partition functions provide combinatorial formulas for cluster variables resulting from mutations of the quiver associated with the del Pezzo surface dP3. In our paper, we investigate a variant of the dP3 quiver by considering a second alphabet of variables that breaks the symmetries of the relevant recurrences. This deformation is motivated by the theory of cluster algebras with principal coefficients introduced by Fomin and Zelevinsky. Our main result gives an explicit formula extending previously known generating functions for dP3 cluster variables by using Aztec castles and constructing their associated minimal matchings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal Matchings for dP3 Cluster Variables Chiang, Judy Hsin-Hui Musiker, Gregg Nguyen, Son Combinatorics 05C70, 06A07, 13F60 In previous work [LM17], Tri Lai and the second author studied a family of subgraphs of the dP3 brane tiling, called Aztec castles, whose dimer partition functions provide combinatorial formulas for cluster variables resulting from mutations of the quiver associated with the del Pezzo surface dP3. In our paper, we investigate a variant of the dP3 quiver by considering a second alphabet of variables that breaks the symmetries of the relevant recurrences. This deformation is motivated by the theory of cluster algebras with principal coefficients introduced by Fomin and Zelevinsky. Our main result gives an explicit formula extending previously known generating functions for dP3 cluster variables by using Aztec castles and constructing their associated minimal matchings. |
| title | Minimal Matchings for dP3 Cluster Variables |
| topic | Combinatorics 05C70, 06A07, 13F60 |
| url | https://arxiv.org/abs/2409.05803 |