Local and global patterns of rank 3 $G$-fans of totally-infinite type

TL;DR

This paper analyzes the structure and patterns of rank 3 $G$-fans of totally-infinite type, classifying local and global behaviors, and demonstrating their incompleteness and new configurations.

Contribution

It provides a classification of local and global patterns of rank 3 $G$-fans of totally-infinite type, revealing their asymptotic behavior and new configurations.

Findings

Classified rank 3 $G$-fans into several patterns.

Proved the incompleteness of $G$-fans of infinite type.

Presented new prototypical examples of global patterns.

Abstract

We focus on the $G$-fans associated with cluster patterns whose initial exchange matrices are of infinite type. We study the asymptotic behavior of the $g$-vectors around the initial $G$-cone under the alternating mutations for two indices of infinite type. In the rank 3 case, we classify them into several patterns. As an application, the incompleteness of the $G$-fans of infinite type is proved. We observed that the local pattern of a rank 3 $G$-fan of totally-infinite type classified by the above types correlates with its global pattern. Following the classification of the local patterns (together with the Markov constant), we present several prototypical examples of the global patterns of the rank 3 $G$-fans of totally-infinite type, many of which are new in the literature.