Explicit forms in lower degrees of rank 2 cluster scattering diagrams

TL;DR

This paper develops a method to explicitly compute and analyze wall elements in rank 2 cluster scattering diagrams, revealing their forms up to degree 7 and identifying degree-independent walls using matrix actions.

Contribution

It introduces a novel method for calculating lower-degree wall elements and their explicit forms, including the Badlands, in rank 2 cluster scattering diagrams.

Findings

Explicit forms of wall elements up to degree 7

Identification of degree-independent walls

Introduction of a matrix similarity transformation

Abstract

In this paper, we study wall elements of rank 2 cluster scattering diagrams based on dilogarithm elements. We derive two major results. First, we give a method to calculate wall elements in lower degrees. By this method, we may see the explicit forms of wall elements including the Badlands, which is the complement of $G$-fan. In this paper, we write one up to 7 degrees. Also, by using this method, we derive some walls independent of their degrees. Second, we find a certain admissible form of them. In the proof of these facts, we introduce a matrix action on a structure group, which we call a similarity transformation, and we argue the relation between this action and ordered products.