2025/09/09 by Ryota Akagi, Zhichao Chen, Akagi, Ryota +1
Computer Science · Mathematics · #Polynomial and algebraic computation #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2509.07454
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 D6 with a cluster realization via certain mutations.