Group actions on relative cluster categories and Higgs categories

TL;DR

This paper constructs G-equivariant relative cluster and Higgs categories for group actions on ice quivers with potential, linking them to skew-symmetrizable cluster algebras with coefficients.

Contribution

It extends Demonet's work by developing G-equivariant categories and connecting orbit mutations to explicit cluster algebras, including non-simply laced cases.

Findings

Constructed G-equivariant relative cluster and Higgs categories.

Linked G-stable cluster-tilting objects to skew-symmetrizable cluster algebras.

Provided an additive categorification for cluster algebras with principal coefficients.

Abstract

Let $G$ be a finite group acting on an ice quiver with potential $(Q, F, W)$. We construct the corresponding $G$-equivariant relative cluster category and $G$-equivariant Higgs category, extending the work of Demonet. Using the orbit mutations on the set of $G$-stable cluster-tilting objects of the Higgs category and an appropriate cluster character, we can link these data to an explicit skew-symmetrizable cluster algebra with coefficients. As a specific example, this provides an additive categorification for cluster algebras with principal coefficients in the non-simply laced case.