Mutation of frozen Jacobian algebras

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.

28 pages, including corrigendum. Earlier version of Section 5 appeared in v1 of arXiv:1702.05352. v2: accepted manuscript, to appear in J. Algebra. v3: added corrigendum

References in corpus (6)

• Dimer models and toric diagrams
• Seiberg Duality for Quiver Gauge Theories
• Deformed Calabi-Yau Completions
• A categorification of acyclic principal coefficient cluster algebras
• Categorification and the quantum Grassmannian
• Quivers with potentials for Grassmannian cluster algebras

Cited by in corpus (8)

• A categorification of acyclic principal coefficient cluster algebras
• Calabi-Yau properties of Postnikov diagrams
• Ginzburg algebras of triangulated surfaces and perverse schobers
• Perfect matching modules, dimer partition functions and cluster characters
• A Lagrangian filling for every cluster seed
• Consistent Dimer Models on Surfaces with Boundary
• Combinatorial mutations of Newton-Okounkov polytopes arising from plabic graphs
• Complexity of quiver mutation equivalence