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arXiv · 1506.04210

A Cambrian framework for the oriented cycle

Nathan Reading·David E. Speyer
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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 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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Nathan Reading, David E. Speyer. 2026-05-27. A Cambrian framework for the oriented cycle. https://doi.org/10.37236/5387

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