Group actions on relative cluster categories and Higgs categories

TL;DR

The paper develops a framework for group actions on ice quivers with potential by constructing $G$-equivariant relative cluster categories and Higgs categories, linking them to skew-symmetrizable cluster algebras with coefficients via orbit mutations and a cluster character. It extends the relative Calabi–Yau and Ginzburg formalisms to the equivariant setting, establishing derived equivalences between skew-group and invariant categories and proving that $G$-stable cluster-tilting data categorifies non-simply-laced cluster algebras with principal coefficients. A key outcome is an additive categorification in the non-simply-laced case, together with a $\mathsf{mod}(k[G])$-linear structure on the equivariant Higgs category and explicit orbit-exchange matrices guiding mutations. The work advances the understanding of how symmetry groups act on categorical models of cluster algebras, enabling systematic construction of categorifications for skew-symmetrizable seeds with coefficients and providing tools for explicit computations via orbit mutations and cluster characters.

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.

Theorems & Definitions (75)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Theorem 2.9
• Definition 3.1