Geometry of mutation classes of rank $3$ quivers

TL;DR

This paper provides a geometric framework for understanding all mutation classes of rank 3 quivers with real weights, linking their behavior to reflection groups and the Markov constant, and classifies certain mutation-finite matrices.

Contribution

It introduces a geometric realization for all rank 3 quiver mutation classes using reflection groups and rotation groups, and classifies mutation-finite real 3x3 matrices.

Findings

Geometric realization via reflection groups and rotations for mutation classes.

Classification of skew-symmetric mutation-finite real 3x3 matrices.

Analysis of acyclic representatives in mutation classes.

Abstract

We present a geometric realization for all mutation classes of quivers of rank $3$ with real weights. This realization is via linear reflection groups for acyclic mutation classes and via groups generated by $\pi$-rotations for the cyclic ones. The geometric behavior of the model turns out to be controlled by the Markov constant $p^2+q^2+r^2-pqr$, where $p,q,r$ are the elements of exchange matrix. We also classify skew-symmetric mutation-finite real $3\times 3$ matrices and explore the structure of acyclic representatives in finite and infinite mutation classes.