A Cambrian framework for the oriented cycle (Q907237)

Summary: This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and \(\mathfrak g\)-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.