Cyclic Sieving for torsion pairs in the cluster category of Dynkin type A_n

TL;DR

This paper explores the cyclic sieving phenomenon in the context of torsion pairs in the cluster category of Dynkin type A_n, extending combinatorial models and explicit formulas to invariant torsion pairs under Auslander-Reiten translation.

Contribution

It determines the count of torsion pairs invariant under multiple Auslander-Reiten translations and demonstrates the cyclic sieving phenomenon with a q-analogue.

Findings

Number of invariant torsion pairs under b-fold Auslander-Reiten translation

Explicit formula for total torsion pairs in Dynkin type A_n

Cyclic sieving phenomenon observed with q-analogue

Abstract

Recently, a combinatorial model for torsion pairs in the cluster category of Dynkin type A_n was introduced, and used to derive an explicit formula for their number. In this article we determine the number of torsion pairs that are invariant under b-fold application of Auslander-Reiten translation. It turns out that the set of torsion pairs together with Auslander-Reiten translation, and a natural q-analogue of the formula for the number of all torsion pairs exhibits the cyclic sieving phenomenon.