A new cluster character with coefficients for cluster category

TL;DR

This paper introduces a new cluster character with coefficients for cluster categories, using intrinsic properties rather than Frobenius 2-Calabi-Yau realizations, and establishes a bijection for Dynkin type A_n.

Contribution

It develops an intrinsic approach to cluster characters with coefficients and links ice quivers to cluster tilting objects in Dynkin type A_n.

Findings

Defined an ice quiver associated to each cluster tilting object.

Proved a bijection between mutation classes of ice quivers and cluster tilting objects in type A_n.

Provided an alternative to Frobenius 2-Calabi-Yau realizations for incorporating coefficients.

Abstract

We introduce a new cluster character with coefficients for a cluster category $\mathcal{C}$ and rather than using a Frobenius $2$-Calabi-Yau realization to incorporate coefficients into the representation-theoretic model for a cluster algebra, as done by Fu and Keller, we exploit intrinsic properties of $\mathcal{C}$. For this purpose, we define an ice quiver associated to each cluster tilting object in $\mathcal{C}$. In Dynkin case $\mathbb{A}_n$, we also prove that the mutation class of the ice quiver associated to the cluster tilting object given by the direct sum of all projective objects is in bijection with set of ice quivers of cluster tilting objects in $\mathcal{C}$.