From green mutation to X-evolution: flows and foliations on cluster complexes

2025/01/27 by Yu Qiu, Qiu, Yu, Tang, Liheng
Economics, Econometrics and Finance · Mathematics · #Complex Systems and Time Series Analysis #FOS: Mathematics #Representation Theory (math.RT) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2501.15756

openalex publication_date 2025/01/27 · openalex created_date 2025/01/29 · openalex updated_date 2026/07/28

Abstract

For any point X in the cluster complex Cpx(C) of a 2-Calabi-Yau category C, we introduce X-evolution flow on Cpx(C). We show that such a flow induces a piecewise linear one-dimensional X-foliation with two singularities, the unique sink X and the unique source X[1]. Moreover, we show that evolution flows on cluster complexes are continuous refinement/generalization of green mutations on cluster exchange graphs. For the cluster category of a Dynkin or Euclidean quiver Q, we prove that the X-foliation is compact or semi-compact, for various choices of X. As an application, we show that Cpx(C) is spherical (Dynkin case) or contractible (Euclidean case). As a byproduct, we show that the fundamental group of the cluster exchange graph of Q is generated by squares and pentagons.

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