Synchronicity phenomenon in cluster patterns
Abstract
It has been known that several objects such as cluster variables, coefficients, seeds, and -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.
Keywords
Cite
@article{arxiv.1906.12036,
title = {Synchronicity phenomenon in cluster patterns},
author = {Tomoki Nakanishi},
journal= {arXiv preprint arXiv:1906.12036},
year = {2024}
}
Comments
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