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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.16326 |
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| _version_ | 1866908893501718528 |
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| author | Akagi, Ryota Chen, Zhichao |
| author_facet | Akagi, Ryota Chen, Zhichao |
| contents | In this paper, we investigate the geometric structures of $G$-fans associated with rank $3$ real cluster-cyclic exchange matrices. In this class, a simple recursion for tropical signs was found, which enables us to study the detailed properties of $c$-, $g$-vectors. We introduce two kinds of upper bounds of the $G$-fans. The first one is the global upper bound, which comes from a hyperbolic surface containing all $g$-vectors after an initial mutation. The second one is the local upper bound, which reflects the internal separateness structure. As applications, we prove that there is no periodicity among $g$-vectors, and we completely determine the sign of $g$-vectors. We also prove the monotonicity of $g$-vectors under the minimum assumption. Moreover, we show that the three global upper bounds can be simplified to a single uniform upper bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16326 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric structures of $G$-fans associated with rank $3$ cluster-cyclic exchange matrices Akagi, Ryota Chen, Zhichao Combinatorics Representation Theory In this paper, we investigate the geometric structures of $G$-fans associated with rank $3$ real cluster-cyclic exchange matrices. In this class, a simple recursion for tropical signs was found, which enables us to study the detailed properties of $c$-, $g$-vectors. We introduce two kinds of upper bounds of the $G$-fans. The first one is the global upper bound, which comes from a hyperbolic surface containing all $g$-vectors after an initial mutation. The second one is the local upper bound, which reflects the internal separateness structure. As applications, we prove that there is no periodicity among $g$-vectors, and we completely determine the sign of $g$-vectors. We also prove the monotonicity of $g$-vectors under the minimum assumption. Moreover, we show that the three global upper bounds can be simplified to a single uniform upper bound. |
| title | Geometric structures of $G$-fans associated with rank $3$ cluster-cyclic exchange matrices |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2603.16326 |