Mutation-classes of diagrams via infinite graphs

TL;DR

This paper classifies the mutation classes of diagrams associated with Dynkin and affine Dynkin types using specific families of diagrams, providing a comprehensive understanding of their structure.

Contribution

It offers a complete description of cluster-mutation classes for Dynkin and affine Dynkin types through new diagram families, advancing the classification theory.

Findings

Complete classification of mutation classes for Dynkin types.

Explicit diagram families representing each class.

Enhanced understanding of cluster algebra structures.

Abstract

We give a complete description of the cluster-mutation classes of diagrams of Dynkin types \mathbb{A},\mathbb{B},\mathbb{D} and of affine Dynkin types \mathbb{B}^{(1)},\mathbb{C}^{(1)},\mathbb{D}^{(1)} via certain families of diagrams.