A Cambrian framework for the oriented cycle

TL;DR

This paper develops a combinatorial framework for exchange graphs and g-vector fans associated with exchange matrices of finite or affine type, extending Cambrian lattice constructions to cyclic orientations.

Contribution

It constructs a Cambrian framework for the oriented n-cycle, generalizing previous acyclic cases and introducing sortable elements for quivers with cycles.

Findings

Constructed a Cambrian framework for the cyclically oriented n-cycle.

Extended the doubled Cambrian fan construction to cyclic orientations.

Unified combinatorial models for finite and affine types with cycles.

Abstract

This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and g-vector fan associated to any exchange matrix B whose Cartan companion is of finite or affine type, using the combinatorics and geometry of Coxeter-sortable elements and Cambrian lattices/fans. Specifically, we construct a framework in the unique non-acyclic affine case, the cyclically oriented n-cycle. In the acyclic affine case, a framework was constructed by combining a copy of the Cambrian fan for B with an antipodal copy of the Cambrian fan for -B. In this paper, we extend this "doubled Cambrian fan" construction to the oriented n-cycle, using a more general notion of sortable elements for quivers with cycles.