272 citations · 1k across the 53 of their papers we have counts for
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Local and global patterns of rank 3 -fans of totally-infinite type
Tomoki Nakanishi
We focus on the -fans associated with cluster patterns whose initial exchange matrices are of infinite type. We study the asymptotic behavior of the -vectors around the initi…
Pentagon relation in quantum cluster scattering diagrams
Tomoki Nakanishi
We formulate the pentagon relation for quantum dilogarithm elements in the structure group of a quantum cluster scattering diagram (QCSD). As an application, we show the nonpositiv…
Cluster Algebras and Scattering Diagrams, Part I. Basics in Cluster Algebras
Tomoki Nakanishi
This is a first step guide to the theory of cluster algebras. We especially focus on basic notions, techniques, and results concerning seeds, cluster patterns, and cluster algebras…
Two Formulas for -Polynomials
Feiyang Lin, Gregg Musiker, Tomoki Nakanishi
We discuss a product formula for -polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for -polynomials. The other…
Dilogarithm identities in cluster scattering diagrams
Tomoki Nakanishi
We extend the notion of -variables (coefficients) in cluster algebras to cluster scattering diagrams. Accordingly, we extend the dilogarithm identity associated with a period in…
Cluster Algebras and Scattering Diagrams, Part III. Cluster Scattering Diagrams
Tomoki Nakanishi
This is a self-contained exposition of several fundamental properties of cluster scattering diagrams introduced and studied by Gross, Hacking, Keel, and Kontsevich. In particular,…