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

Abstract

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.

In this version, we improved the exposition and the structure throughout the paper (especially the Sections 4, 6, 7 in the last version). We also added some explanations and references concerning related works. Moreover, we simplified several unnecessary proofs and made the statements more clear (especially about Proposition 3.7, Proposition 4.13 and Proposition 6.1 in the present version)