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arXiv · 2603.16326

Geometric structures of $G$-fans associated with rank $3$ cluster-cyclic exchange matrices

Ryota Akagi·Zhichao Chen
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Abstract

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.

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Ryota Akagi, Zhichao Chen. 2026-03-17. Geometric structures of $G$-fans associated with rank $3$ cluster-cyclic exchange matrices. https://arxiv.org/abs/2603.16326

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