Ptolemy diagrams and torsion pairs in the cluster categories of Dynkin type D

TL;DR

This paper classifies torsion pairs in the cluster category of Dynkin type D_n using Ptolemy diagrams, introduces combinatorial descriptions, and counts their number via an explicit generating function.

Contribution

It provides a complete classification of torsion pairs in type D_n cluster categories through novel Ptolemy diagrams and combinatorial descriptions, including an explicit counting method.

Findings

Classified torsion pairs in type D_n cluster categories.

Introduced Ptolemy diagrams of type D as combinatorial models.

Derived an explicit generating function for counting torsion pairs.

Abstract

We give a complete classification of torsion pairs in the cluster category of Dynkin type D_n, via a bijection to new combinatorial objects called Ptolemy diagrams of type D. For the latter we give along the way different combinatorial descriptions. One of these allows us to count the number of torsion pairs in the cluster category of type D_n by providing their generating function explicitly.