Cluster-cyclic condition of skew-symmetrizable matrices of rank 3 via the Markov constant

TL;DR

This paper investigates the conditions under which mutations of skew-symmetrizable matrices of rank 3 always produce cyclic quivers, emphasizing the role of the Markov constant in this process.

Contribution

It introduces a new criterion based on the Markov constant to determine when mutated quivers remain cyclic for skew-symmetrizable matrices of rank 3.

Findings

Identifies conditions for cyclic quivers after mutations

Highlights the significance of the Markov constant in matrix mutations

Provides a framework for analyzing mutations of skew-symmetrizable matrices

Abstract

In this paper, we consider mutations of skew-symmetrizable matrices of rank 3. Every skew-symmetrizable matrix corresponds to a weighted quiver, and we study the conditions when this quiver is always cyclic after applying mutations. In this study, the Markov constant has an essential meaning. It has already appeared in some previous works for skew-symmetric matrices.