A note on the spectral properties of cluster algebras

TL;DR

This paper explores the spectral properties of quivers related to cluster algebras and their connection to the representation theory of path algebras, focusing on mutation equivalence.

Contribution

It establishes a link between cluster algebra spectral properties and the representation theory of path algebras for quivers without oriented cycles.

Findings

Spectral properties are characterized for quivers mutation equivalent to a given acyclic quiver.

Connection between spectral graph theory and cluster algebra mutations is demonstrated.

Provides new insights into the structure of quivers in relation to their spectral data.

Abstract

Let $Q$ be a finite quiver without oriented cycles and $k$ an algebraically closed field.In this paper we establish a connection between cluster algebras and the representation theory of the path algebra $kQ$, in terms of the spectral properties of the quivers mutation equivalent to $Q$.