On the Number of Arrows of Cluster Quivers

2021/05/24 by Qiuning Du, Fang Li, Du, Qiuning +3
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Cluster (spacecraft) #Cluster algebra #Combinatorics #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Computer science #Discrete mathematics #Distribution (mathematics) #FOS: Mathematics #Finite set #Mathematical analysis #Mathematics #Mutation #Physics #Pure mathematics #Quiver #Representation Theory (math.RT) #Set (abstract data type) #Statistical physics #Type (biology) #math.AC #math.CO #math.RT

paper · pdf · doi:10.48550/arxiv.2105.11278

published in arXiv (Cornell University) (Cornell University)

arxiv created 2021/05/24 · openalex publication_date 2021/05/24 · arxiv updated 2021/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let Q (resp. Q) be an extended exchange (resp. exchange) cluster quiver of finite mutation type. We introduce the distribution set of the number of arrows for Mut[Q] (resp. Mut[Q]), give the maximum and minimum numbers of the distribution set and establish the existence of an extended complete walk (resp. a complete walk). As a consequence, we prove that the distribution set for Mut[Q] (resp. Mut[Q]) is continuous except the exceptional cases. In case of cluster quivers Q_inf of infinite mutation type, the number of arrows does not present a continuous distribution. Besides, we show that the maximal number of arrows of quivers in Mut[Q_inf] is infinite if and only if the maximal number of arrows between any two vertices of a quiver in Mut[Q_inf] is infinite.

Citations

• Dual Teichmuller and lamination spaces

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