Mutation classes of finite type cluster algebras with principal coefficients

TL;DR

This paper proves a conjecture that characterizes finite mutation classes of matrices in finite type cluster algebras, confirming they are exactly those with finite classes, thus advancing understanding of cluster algebra structure.

Contribution

It confirms a key conjecture by Fomin and Zelevinsky, precisely identifying finite mutation classes in finite type cluster algebras.

Findings

Proves Conjecture 4.8 from 'Cluster algebras IV'

Characterizes finite mutation classes as exactly those that are finite

Advances classification of cluster algebra mutation classes

Abstract

In this paper, we prove Conjecture 4.8 of "Cluster algebras IV" by S. Fomin and A. Zelevinsky, stating that the mutation classes of rectangular matrices associated with cluster algebras of finite type are precisely those classes which are finite.