Real $C$-, $G$-structures and sign-coherence of cluster algebras

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 the one of skew-symmetric patterns. In this sense, we extend the sign-coherence of integer $C$-, $G$-matrices proved by Gross-Hacking-Keel-Kontsevich to a more general real class called of quasi-integer type. Furthermore, we give a complete classification of this type by a combinatorial method of real weighted quivers. 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.