Machine Learning Mutation-Acyclicity of Quivers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915487843090432 |
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| author | Armstrong-Williams, Kymani T. K. Hirst, Edward Jackson, Blake Lee, Kyu-Hwan |
| author_facet | Armstrong-Williams, Kymani T. K. Hirst, Edward Jackson, Blake Lee, Kyu-Hwan |
| contents | Machine learning (ML) has emerged as a powerful tool in mathematical research in recent years. This paper applies ML techniques to the study of quivers -- a type of directed multigraph with significant relevance in algebra, combinatorics, computer science, and mathematical physics. Specifically, we focus on the challenging problem of determining the mutation-acyclicity of a quiver on 4 vertices, a property that is pivotal since mutation-acyclicity is often a necessary condition for theorems involving path algebras and cluster algebras. Although this classification is known for quivers with at most 3 vertices, little is known about quivers on more than 3 vertices. We give a computer-assisted proof of a theorem to prove that mutation-acyclicity is decidable for quivers on 4 vertices with edge weight at most 2. By leveraging neural networks (NNs) and support vector machines (SVMs), we then accurately classify more general 4-vertex quivers as mutation-acyclic or non-mutation-acyclic. Our results demonstrate that ML models can efficiently detect mutation-acyclicity, providing a promising computational approach to this combinatorial problem, from which the trained SVM equation provides a starting point to guide future theoretical development. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Machine Learning Mutation-Acyclicity of Quivers Armstrong-Williams, Kymani T. K. Hirst, Edward Jackson, Blake Lee, Kyu-Hwan Combinatorics Machine Learning High Energy Physics - Theory Representation Theory 13F60 (Primary) 05-08, 68T07, 68V05 (Secondary) G.2.1; I.2.6; J.2 Machine learning (ML) has emerged as a powerful tool in mathematical research in recent years. This paper applies ML techniques to the study of quivers -- a type of directed multigraph with significant relevance in algebra, combinatorics, computer science, and mathematical physics. Specifically, we focus on the challenging problem of determining the mutation-acyclicity of a quiver on 4 vertices, a property that is pivotal since mutation-acyclicity is often a necessary condition for theorems involving path algebras and cluster algebras. Although this classification is known for quivers with at most 3 vertices, little is known about quivers on more than 3 vertices. We give a computer-assisted proof of a theorem to prove that mutation-acyclicity is decidable for quivers on 4 vertices with edge weight at most 2. By leveraging neural networks (NNs) and support vector machines (SVMs), we then accurately classify more general 4-vertex quivers as mutation-acyclic or non-mutation-acyclic. Our results demonstrate that ML models can efficiently detect mutation-acyclicity, providing a promising computational approach to this combinatorial problem, from which the trained SVM equation provides a starting point to guide future theoretical development. |
| title | Machine Learning Mutation-Acyclicity of Quivers |
| topic | Combinatorics Machine Learning High Energy Physics - Theory Representation Theory 13F60 (Primary) 05-08, 68T07, 68V05 (Secondary) G.2.1; I.2.6; J.2 |
| url | https://arxiv.org/abs/2411.04209 |