An unexpected property of $\mathbf{g}$-vectors for rank 3 mutation-cyclic quivers
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910586128826368 |
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| author | Lee, Jihyun Lee, Kyungyong |
| author_facet | Lee, Jihyun Lee, Kyungyong |
| contents | Let $Q$ be a rank 3 mutation-cyclic quiver. It is known that every $\mathbf{c}$-vector of $Q$ is a solution to a quadratic equation of the form $$\sum_{i=1}^3 x_i^2 + \sum_{1\leq i<j\leq 3} \pm q_{ij} x_i x_j =1,$$where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$. A similar property holds for $\mathbf{c}$-vectors of any acyclic quiver. In this paper, we show that $\mathbf{g}$-vectors of $Q$ enjoy an unexpected property. More precisely, every $\mathbf{g}$-vector of $Q$ is a solution to a quadratic equation of the form $$\sum_{i=1}^3 x_i^2 + \sum_{1\leq i<j\leq 3} p_{ij} x_i x_j =1,$$where $p_{ij}$ is the number of arrows between the vertices $i$ and $j$ in another quiver $P$ obtained by mutating $Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00599 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An unexpected property of $\mathbf{g}$-vectors for rank 3 mutation-cyclic quivers Lee, Jihyun Lee, Kyungyong Combinatorics Let $Q$ be a rank 3 mutation-cyclic quiver. It is known that every $\mathbf{c}$-vector of $Q$ is a solution to a quadratic equation of the form $$\sum_{i=1}^3 x_i^2 + \sum_{1\leq i<j\leq 3} \pm q_{ij} x_i x_j =1,$$where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$. A similar property holds for $\mathbf{c}$-vectors of any acyclic quiver. In this paper, we show that $\mathbf{g}$-vectors of $Q$ enjoy an unexpected property. More precisely, every $\mathbf{g}$-vector of $Q$ is a solution to a quadratic equation of the form $$\sum_{i=1}^3 x_i^2 + \sum_{1\leq i<j\leq 3} p_{ij} x_i x_j =1,$$where $p_{ij}$ is the number of arrows between the vertices $i$ and $j$ in another quiver $P$ obtained by mutating $Q$. |
| title | An unexpected property of $\mathbf{g}$-vectors for rank 3 mutation-cyclic quivers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.00599 |