A Combinatorial Characterization of Cluster Algebras: On the Number of Arrows of Cluster Quivers

Abstract

Let $\tilde{Q}$ (resp. Q) be an extended exchange (resp. exchange) cluster quiver of finite mutation type. We introduce the distribution set of the numbers of arrows for $Mut[\tilde{Q}]$ (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 $Mut[\tilde{Q}]$ (resp. Mut[Q]) is continuous except in the case of exceptional cluster algebras. In case of cluster quivers $Q_{inf}$ of infinite mutation type, the distribution set for $Mut[Q_{inf}]$ in general is not continuous. 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.

6. Appendix

In this appendix we show that when $n=4$, iteratively applying mutations $\mu _{3}\mu _{2}\mu _{1}$ always changes the quiver to $Q^{*}$. And during this process the multiplicities of a, b, c never decrease.