Mutation-finite quivers with real weights

TL;DR

This paper classifies all mutation-finite quivers with real weights, revealing their geometric realizations and exploring the structure of acyclic representatives within finite mutation classes.

Contribution

It provides a complete classification of mutation-finite quivers with real weights and links their structure to geometric reflection groups.

Findings

All mutation-finite quivers with real weights are classified.

Finite mutation classes not from integer matrices have geometric realizations.

Acyclic representatives relate to acute-angled simplicial domains.

Abstract

We classify all mutation-finite quivers with real weights. We show that every finite mutation class not originating from an integer skew-symmetrizable matrix has a geometric realization by reflections. We also explore the structure of acyclic representatives in finite mutation classes and their relations to acute-angled simplicial domains in the corresponding reflection groups.