Torsion pairs in cluster tubes

TL;DR

This paper classifies torsion pairs in cluster categories of tubes using combinatorial Ptolemy diagrams, providing enumeration formulas and revealing a cyclic sieving phenomenon.

Contribution

It offers a complete classification of torsion pairs in cluster tubes and connects them to Ptolemy diagrams, extending previous work on Dynkin type A.

Findings

Classified all torsion pairs in cluster tubes

Derived closed-form enumeration formulas

Discovered cyclic sieving phenomenon in torsion pairs

Abstract

We give a complete classification of torsion pairs in the cluster categories associated to tubes of finite rank. The classification is in terms of combinatorial objects called Ptolemy diagrams which already appeared in our earlier work on torsion pairs in cluster categories of Dynkin type A. As a consequence of our classification we establish closed formulae enumerating the torsion pairs in cluster tubes, and obtain that the torsion pairs in cluster tubes exhibit a cyclic sieving phenomenon.