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Denominator vectors and compatibility degrees in cluster algebras of finite type


Authors: Cesar Ceballos and Vincent Pilaud
Journal: Trans. Amer. Math. Soc. 367 (2015), 1421-1439
MSC (2010): Primary 13F60; Secondary 20F55, 05E15, 05E45
Published electronically: August 12, 2014
MathSciNet review: 3280049
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Abstract: We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra. They provide two simple proofs of the known fact that the $ d$-vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero.


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Additional Information

Cesar Ceballos
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3
Email: ceballos@mathstat.yorku.ca

Vincent Pilaud
Affiliation: CNRS & LIX, École Polytechnique, Palaiseau, France
Email: vincent.pilaud@lix.polytechnique.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06239-9
Keywords: Cluster algebras, subword complexes, denominator vectors, compatibility degrees
Received by editor(s): May 27, 2013
Received by editor(s) in revised form: July 8, 2013
Published electronically: August 12, 2014
Additional Notes: The first author was supported by DFG via the Research Training Group “Methods for Discrete Structures” and the Berlin Mathematical School. He was also partially supported by the government of Canada through a Banting Postdoctoral Fellowship.
The second author was supported by the Spanish MICINN grant MTM2011-22792, by the French ANR grant EGOS 12 JS02 002 01, and by the European Research Project ExploreMaps (ERC StG 208471).
Article copyright: © Copyright 2014 American Mathematical Society