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

TL;DR

This paper explores the geometric structures of $G$-fans linked to rank 3 cluster-cyclic exchange matrices, introducing bounds and properties of $g$-vectors, with implications for their periodicity and sign patterns.

Contribution

It introduces new upper bounds for $G$-fans and provides detailed analysis of $g$-vector properties, including non-periodicity and sign determination, in the context of rank 3 cluster-cyclic matrices.

Findings

No periodicity among $g$-vectors.

Complete sign determination of $g$-vectors.

Simplification of global upper bounds to a single bound.

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.