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

Ryota Akagi, Zhichao Chen

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.

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

Abstract

In this paper, we investigate the geometric structures of -fans associated with rank 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 -, -vectors. We introduce two kinds of upper bounds of the -fans. The first one is the global upper bound, which comes from a hyperbolic surface containing all -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 -vectors, and we completely determine the sign of -vectors. We also prove the monotonicity of -vectors under the minimum assumption. Moreover, we show that the three global upper bounds can be simplified to a single uniform upper bound.
Paper Structure (39 sections, 61 theorems, 233 equations, 10 figures, 2 tables)

This paper contains 39 sections, 61 theorems, 233 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Purpose and related works
  4. Conventions
  5. Main results
  6. Structure of this paper
  7. Preliminaries
  8. $B$-, $C$-, $G$-matrices
  9. $G$-fan structure
  10. Skew-symmetrizing method
  11. Modified $c$-, $g$-vectors
  12. Stereographic projection
  13. Rank $3$ cluster-cyclic framework
  14. Markov constant and cluster-cyclicity
  15. Mutation of tropical signs
  16. ...and 24 more sections

Key Result

Theorem 1.1

The following inclusion holds: In particular, we have $|\Delta(B)| \subset Q_1^+ \cup Q_2^+ \cup Q_3^+$.

Figures (10)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 6:
  • ...and 5 more figures

Theorems & Definitions (139)

  • Theorem 1.1: \ref{['thm: global upper bound theorem']}
  • Theorem 1.2: \ref{['thm: local upper bound theorem']}
  • Theorem 1.3: \ref{['thm: non periodicity']}
  • Theorem 1.4: \ref{['thm: sign of G matrix']}
  • Theorem 1.5: \ref{['thm: monotonicity of g-vectors']}
  • Theorem 1.6: \ref{['thm: simplified global upper bound']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 129 more