2025/09/08 by Ryota Akagi, Zhichao Chen, Akagi, Ryota +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2509.06486
We generalize the theory of integer C-, G-matrices in cluster algebras to the real case. By a skew-symmetrizing method, we can reduce the problem of skew-symmetrizable patterns to skew-symmetric patterns. In this sense, the sign-coherence of a more general real class called of quasi-integer type can be inherited directly from that of integer C-, G-matrices proved by Gross-Hacking-Keel-Kontsevich. However, the sign-coherence of real C-, G-matrices does not always hold in general. For this purpose, we classify all the rank 2 case and the finite type case via the Coxeter diagrams. We also give two conjectures about the real exchange matrices and C-, G-matrices. Under these conjectures, the dual mutation, G-fan structure and synchronicity property hold. As an application, the isomorphism of several kinds of exchange graphs is studied.