Subquivers of mutation-acyclic quivers are mutation-acyclic

TL;DR

This paper proves that any full subquiver of a mutation-acyclic quiver is also mutation-acyclic, enhancing understanding of the structural properties preserved under quiver mutation in cluster algebras.

Contribution

It provides a proof that mutation-acyclicity is preserved when passing to full subquivers, a previously unresolved property in the theory of quiver mutations.

Findings

Full subquivers of mutation-acyclic quivers are mutation-acyclic

Supports structural stability in quiver mutation theory

Enhances understanding of cluster algebra properties

Abstract

Quiver mutation plays a crucial role in the definition of cluster algebras by Fomin and Zelevinsky. It induces an equivalence relation on the set of all quivers without loops and two-cycles. A quiver is called mutation-acyclic if it is mutation-equivalent to an acyclic quiver. This note gives a proof that full subquivers of mutation-acyclic quivers are mutation-acyclic.