Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A_n

TL;DR

This paper classifies torsion pairs in the cluster category of Dynkin type A_n, introduces a new combinatorial description of Ptolemy diagrams, and counts these pairs including their equivalence under Auslander-Reiten translation.

Contribution

It provides a complete classification of torsion pairs and a novel combinatorial framework for Ptolemy diagrams in the context of Dynkin type A_n.

Findings

Complete classification of torsion pairs in the cluster category of type A_n

New combinatorial description of Ptolemy diagrams

Counting of torsion pairs and their equivalence classes

Abstract

We give a complete classification of torsion pairs in the cluster category of Dynkin type A_n. Along the way we give a new combinatorial description of Ptolemy diagrams, an infinite version of which was introduced by Ng. This allows us to count the number of torsion pairs in the cluster category of type A_n. We also count torsion pairs up to Auslander-Reiten translation.