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Cambrian frameworks for cluster algebras of affine type


Authors: Nathan Reading and David E. Speyer
Journal: Trans. Amer. Math. Soc. 370 (2018), 1429-1468
MSC (2010): Primary 13F60, 20F55
DOI: https://doi.org/10.1090/tran/7193
Published electronically: September 15, 2017
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Abstract: We give a combinatorial model for the exchange graph and $ \mathbf {g}$-vector fan associated to any acyclic exchange matrix $ B$ of affine type. More specifically, we construct a reflection framework for $ B$ in the sense of [N. Reading and D. E. Speyer, ``Combinatorial frameworks for cluster algebras''] and establish good properties of this framework. The framework (and in particular the $ \mathbf {g}$-vector fan) is constructed by combining a copy of the Cambrian fan for $ B$ with an antipodal copy of the Cambrian fan for $ -B$.


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Additional Information

Nathan Reading
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
Email: reading@math.ncsu.edu

David E. Speyer
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
Email: speyer@umich.edu

DOI: https://doi.org/10.1090/tran/7193
Received by editor(s): December 30, 2015
Received by editor(s) in revised form: January 23, 2017
Published electronically: September 15, 2017
Additional Notes: The first author was partially supported by NSA grant H98230-09-1-0056, by Simons Foundation grant #209288, and by NSF grant DMS-1101568. The second author was supported in part by a Clay Research Fellowship and by NSF grant DMS-1600223.
Article copyright: © Copyright 2017 by Nathan Reading and David E. Speyer

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