Orbifold diagrams

TL;DR

This paper introduces orbifold diagrams, a class of diagrams derived from Postnikov diagrams with orbifold points, and establishes their associated quivers with potentials, linking their Jacobian algebras through skew-group algebra constructions.

Contribution

It extends the theory of Postnikov diagrams by defining orbifold diagrams and constructs their quivers with potentials, connecting their Jacobian algebras via skew-group algebra methods.

Findings

Associated quivers with potentials are constructed for orbifold diagrams.

Jacobian algebras are related through skew-group algebra constructions.

Realization of Jacobian algebra as endomorphism algebra of a cluster-tilting object.

Abstract

We study alternating strand diagrams on the disk with an orbifold point. These are quotients by rotation of Postnikov diagrams on the disk, and we call them orbifold diagrams. We associate a quiver with potential to each orbifold diagram, in such a way that its Jacobian algebra and the one associated to the covering Postnikov diagram are related by a skew-group algebra construction. We moreover realise this Jacobian algebra as the endomorphism algebra of a certain explicit cluster-tilting object. This is similar to (and relies on) a result by Baur-King-Marsh for Postnikov diagrams on the disk.