A Cambrian Framework for the Oriented Cycle

  • Nathan Reading
  • David E Speyer
Keywords: Cluster algebra, Framework, Affine root system


This paper completes the project of constructing combinatorial models (called frameworks) for the exchange graph and $\mathbf{g}$-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.

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