Synchronicity phenomenon in cluster patterns

Abstract

It has been known that several objects such as cluster variables, coefficients, seeds, and $Y$-seeds in different cluster patterns with common exchange matrices share the same periodicity under mutations. We call it synchronicity phenomenon in cluster patterns. In this expository note we explain the mechanism of synchronicity based on several fundamental results on cluster algebra theory such as separation formulas, sign-coherence, Laurent positivity, duality, and detropicalization obtained by several authors. We also show that all synchronicity properties studied in this paper are naturally extended to cluster patterns of generalized cluster algebras, up to the Laurent positivity conjecture.

v1: 21pages; v2: 22 pages, references added, minor changes, v3: 23 pages, Sec. 4.1, 5.1, 5.2 revised, Sec.5.6 added, +minor changes, v4: 39 pages, major revision, Part II for generalized cluster algebras added

References in corpus (2)

• The quantum dilogarithm and representations quantum cluster varieties
• The enough $g$-pairs property and denominator vectors of cluster algebras

Cited by in corpus (6)

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• Cluster realizations of Weyl groups and higher Teichmüller theory
• Earthquake theorem for cluster algebras of finite type
• Deformation Theory for Finite Cluster Complexes
• Discrete Dynamical Systems From Real Valued Mutation
• Unbounded ${\mathfrak{sl}}_3$-laminations and their shear coordinates