Categorification of skew-symmetrizable cluster algebras

TL;DR

This paper introduces a new categorification framework for skew-symmetrizable cluster algebras using group actions on 2-Calabi-Yau categories, enabling explicit constructions and proofs of linear independence.

Contribution

It develops a G-equivariant mutation theory in 2-Calabi-Yau categories and constructs skew-symmetrizable cluster algebras from these data, extending previous results to non simply-laced cases.

Findings

Proved linear independence of cluster monomials.

Constructed explicit skew-symmetrizable cluster algebras from categorical data.

Generalized results to partial flag varieties and unipotent subgroups.

Abstract

We propose a new framework for categorifying skew-symmetrizable cluster algebras. Starting from an exact stably 2-Calabi-Yau category C endowed with the action of a finite group G, we construct a G-equivariant mutation on the set of maximal rigid G-invariant objects of C. Using an appropriate cluster character, we can then attach to these data an explicit skew-symmetrizable cluster algebra. As an application we prove the linear independence of the cluster monomials in this setting. Finally, we illustrate our construction with examples associated with partial flag varieties and unipotent subgroups of Kac-Moody groups, generalizing to the non simply-laced case several results of Gei\ss-Leclerc-Schr\"oer.