Mutation of frozen Jacobian algebras

TL;DR

This paper explores mutations of Jacobian algebras, extending to frozen Jacobian algebras from dimer models and cluster categorifications, demonstrating compatibility with cluster algebra mutations and introducing new combinatorial mutation rules.

Contribution

It extends mutation theory to frozen Jacobian algebras and shows compatibility with cluster-tilting mutations in categorifications, including new mutation rules for frozen vertices.

Findings

Mutation of cluster-tilting objects aligns with Fomin-Zelevinsky quiver mutations.

Extension of mutation rules to include arrows between frozen vertices.

Application to categorifications of cluster algebras with frozen variables.

Abstract

We survey results on mutations of Jacobian algebras, while simultaneously extending them to the more general setup of frozen Jacobian algebras, which arise naturally from dimer models with boundary and in the context of the additive categorification of cluster algebras with frozen variables via Frobenius categories. As an application, we show that the mutation of cluster-tilting objects in various such categorifications, such as the Grassmannian cluster categories of Jensen-King-Su, is compatible with Fomin-Zelevinsky mutation of quivers. We also describe an extension of this combinatorial mutation rule allowing for arrows between frozen vertices, which the quivers arising from categorifications and dimer models typically have.