@MISC{Reading15acambrian, author = {Nathan Reading and David E Speyer}, title = {A Cambrian framework for the oriented cycle}, year = {2015} }
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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 ori-ented 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.