Group actions on relative cluster categories and Higgs categories

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.

arXiv comment36 pages, comments are welcome; adding Remark 3.22 and references; to appear in Nagoya Math. J