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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Denominator vectors and compatibility degrees in cluster algebras of finite type
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by Cesar Ceballos and Vincent Pilaud PDF
Trans. Amer. Math. Soc. 367 (2015), 1421-1439 Request permission


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
  • MR Author ID: 990795
  • Email:
  • Vincent Pilaud
  • Affiliation: CNRS & LIX, École Polytechnique, Palaiseau, France
  • MR Author ID: 860480
  • Email:
  • 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).
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 367 (2015), 1421-1439
  • MSC (2010): Primary 13F60; Secondary 20F55, 05E15, 05E45
  • DOI:
  • MathSciNet review: 3280049