Mutations of quivers with 2-cycles
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915709364207616 |
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| author | Li, Fang Liu, Siyang Mou, Lang Pan, Jie |
| author_facet | Li, Fang Liu, Siyang Mou, Lang Pan, Jie |
| contents | We develop a mutation theory for quivers with oriented 2-cycles using a structure called a homotopy, defined as a normal subgroupoid of the quiver's fundamental groupoid. This framework extends Fomin-Zelevinsky mutations of 2-acyclic quivers and yields involutive mutations that preserve the fundamental groupoid quotient by the homotopy. It generalizes orbit mutations arising from quiver coverings and allows for infinite mutation sequences even when orbit mutations are obstructed. We further construct quivers with homotopies from triangulations of marked surfaces with colored punctures, and prove that flips correspond to mutations, extending the Fomin-Shapiro-Thurston model to the setting with 2-cycles. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_15022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mutations of quivers with 2-cycles Li, Fang Liu, Siyang Mou, Lang Pan, Jie Combinatorics Representation Theory 13F60, 16G20, 55P10 We develop a mutation theory for quivers with oriented 2-cycles using a structure called a homotopy, defined as a normal subgroupoid of the quiver's fundamental groupoid. This framework extends Fomin-Zelevinsky mutations of 2-acyclic quivers and yields involutive mutations that preserve the fundamental groupoid quotient by the homotopy. It generalizes orbit mutations arising from quiver coverings and allows for infinite mutation sequences even when orbit mutations are obstructed. We further construct quivers with homotopies from triangulations of marked surfaces with colored punctures, and prove that flips correspond to mutations, extending the Fomin-Shapiro-Thurston model to the setting with 2-cycles. |
| title | Mutations of quivers with 2-cycles |
| topic | Combinatorics Representation Theory 13F60, 16G20, 55P10 |
| url | https://arxiv.org/abs/2508.15022 |