Graded mutation in cluster categories coming from hereditary categories with a tilting object

TL;DR

This paper introduces a graded mutation rule for quivers of cluster-tilted algebras and provides a method to recover cluster-tilting objects from their graded quivers within the cluster category of coherent sheaves.

Contribution

It presents a novel graded mutation rule and a technique to reconstruct cluster-tilting objects from graded quivers in specific cluster categories.

Findings

Established a graded mutation rule for quivers of cluster-tilted algebras.

Developed a method to recover cluster-tilting objects from graded quivers.

Applied the techniques within the cluster category of coherent sheaves.

Abstract

We present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique to recover a cluster-tilting object from its graded quiver in the cluster category of coh $\mathbb{X}$.