| Abstract: |
| We study the sign-coherence of $c$-vectors for rank $3$ real cluster-cyclic skew-symmetrizable cluster algebras. The sign-coherence property was conjectured by Fomin-Zelevinsky and proved in the integer skew-symmetrizable case by Gross-Hacking-Keel-Kontsevich. We extend this result to the rank $3$ real cluster-cyclic setting. In addition, we establish a self-contained recursion and a monotonicity property for these $c$-vectors, and show that they arise as roots of certain quadratic equations. As applications, we prove that the exchange graphs of the associated $C$-patterns and $G$-patterns are $3$-regular trees. We also investigate tropical sign patterns and realize the dihedral group $\mathcal{D}_6$ via cluster mutations. This is a joint work with Ryota Akagi. |
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