Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams

We review some important results by Gross, Hacking, Keel, and Kontsevich on cluster algebra theory, namely, the column sign-coherence of $C$-matrices and the Laurent positivity, both of which were conjectured by Fomin and Zelevinsky. We digest and reconstruct the proofs of these conjectures by Gross et al. still based on their scattering diagram method, however, without relying on toric geometry. At the same time, we also give a detailed account of the correspondence between the notions of cluster patterns and scattering diagrams. Most of the results in this text are found in or translated from the known results in the literature. However, the approach, the construction of logic and proofs, and the overall presentation are new. Also, as an application of the results and the techniques in the text, we show that there is a one-to-one correspondence between $g$-vectors and cluster variables in cluster patterns with arbitrary coefficients.