Sign-coherence and tropical sign pattern for rank $3$ real cluster-cyclic exchange matrices

TL;DR

This work extends sign-coherence and tropical sign theory to rank-3 real cluster-cyclic exchange matrices, establishing a robust recursion for tropical signs and monotonicity of $c$-vectors, and showing these vectors satisfy quadratic equations via a real quasi-Cartan congruence. It proves that the exchange graphs for $C$- and $G$-patterns are $3$-regular trees and uncovers a detailed structure of tropical signs, including trunk/branch dynamics that yield a fractal, dihedral-group–driven mutation theory. A geometric model based on polygons is developed, providing a cluster realization of the dihedral group $ ext{D}_6$ and a concrete visualization of tropical signs under mutations. Collectively, these results enhance the understanding of real rank-3 cluster algebras by linking sign-coherence, quadratic relations, graph structure, and group actions into a cohesive framework.

Abstract

The sign-coherence about $c$-vectors was conjectured by Fomin-Zelevinsky and solved completely by Gross-Hacking-Keel-Kontsevich for integer skew-symmetrizable case. We prove this conjecture associated with $c$-vectors for rank 3 real cluster-cyclic skew-symmetrizable case. Simultaneously, we establish their self-contained recursion and monotonicity. Then, these $c$-vectors are proved to be roots of certain quadratic equations. Based on these results, we prove that the corresponding exchange graphs of $C$-pattern and $G$-pattern are $3$-regular trees. We also study the structure of tropical signs and equip the dihedral group $\mathrm{D}_6$ with a cluster realization via certain mutations.

Figures (7)

• Figure 1: Tropical signs for real cluster-cyclic matrices
• Figure 2: The base case for $r=2$
• Figure 3: Proof of Lemma \ref{['lem: set E']} and Theorem \ref{['thm: group structure of M']}
• Figure 4: $\mathcal{E}^{\geq {\bf w}_0}$
• Figure 5: $\mathcal{E}^{\geq {\bf w}_1}$

Theorems & Definitions (85)

• Theorem 1.2: \ref{['thm: sign-coherency']}
• Theorem 1.3: \ref{['thm: recursion for tropical signs']}
• Theorem 1.4: \ref{['duality']}
• Theorem 1.5: \ref{['thm: exchange graph C']}
• Corollary 1.6: \ref{['cor: exchange graph']}
• Theorem 1.7: \ref{['thm: group structure of M']}
• Definition 2.1: Real exchange matrix, $C,G$-matrices
• Remark 2.2
• Proposition 2.3: Nak23
• Definition 2.4: $\varepsilon$-pattern