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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.08510 |
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| _version_ | 1866917800473264128 |
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| author | Ervin, Tucker J. Jackson, Blake Lee, Kyungyong Nguyen, Son Dang |
| author_facet | Ervin, Tucker J. Jackson, Blake Lee, Kyungyong Nguyen, Son Dang |
| contents | The set of forks is a class of quivers introduced by M. Warkentin, where every connected mutation-infinite quiver is mutation equivalent to infinitely many forks. Let $Q$ be a fork with $n$ vertices, and $\boldsymbol{w}$ be a fork-preserving mutation sequence. We show that every $c$-vector of $Q$ obtained from $\boldsymbol{w}$ is a solution to a quadratic equation of the form $$\sum_{i=1}^n x_i^2 + \sum_{1\leq i<j\leq n} \pm q_{ij} x_i x_j =1,$$ where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$. The same proof techniques implies that when $Q$ is a rank 3 mutation-cyclic quiver, every $c$-vector of $Q$ is a solution to a quadratic equation of the same form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08510 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometry of $C$-vectors and $C$-Matrices for Mutation-Infinite Quivers Ervin, Tucker J. Jackson, Blake Lee, Kyungyong Nguyen, Son Dang Combinatorics The set of forks is a class of quivers introduced by M. Warkentin, where every connected mutation-infinite quiver is mutation equivalent to infinitely many forks. Let $Q$ be a fork with $n$ vertices, and $\boldsymbol{w}$ be a fork-preserving mutation sequence. We show that every $c$-vector of $Q$ obtained from $\boldsymbol{w}$ is a solution to a quadratic equation of the form $$\sum_{i=1}^n x_i^2 + \sum_{1\leq i<j\leq n} \pm q_{ij} x_i x_j =1,$$ where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$. The same proof techniques implies that when $Q$ is a rank 3 mutation-cyclic quiver, every $c$-vector of $Q$ is a solution to a quadratic equation of the same form. |
| title | Geometry of $C$-vectors and $C$-Matrices for Mutation-Infinite Quivers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.08510 |