Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-2014-06239-9
Published electronically: August 12, 2014
MathSciNet review: 3280049
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [BMR07] Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), no. 1, 323-332 (electronic). MR 2247893 (2007f:16035), https://doi.org/10.1090/S0002-9947-06-03879-7
  • [CCS06] Philippe Caldero, Frédéric Chapoton, and Ralf Schiffler, Quivers with relations and cluster tilted algebras, Algebr. Represent. Theory 9 (2006), no. 4, 359-376. MR 2250652 (2007f:16036), https://doi.org/10.1007/s10468-006-9018-1
  • [CLS13] Cesar Ceballos, Jean-Philippe Labbé, and Christian Stump, Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebraic Combin. 39 (2014), no. 1, 17-51. MR 3144391, https://doi.org/10.1007/s10801-013-0437-x
  • [FZ02] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529 (electronic). MR 1887642 (2003f:16050), https://doi.org/10.1090/S0894-0347-01-00385-X
  • [FZ03a] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63-121. MR 2004457 (2004m:17011), https://doi.org/10.1007/s00222-003-0302-y
  • [FZ03b] Sergey Fomin and Andrei Zelevinsky, $ Y$-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018. MR 2031858 (2004m:17010), https://doi.org/10.4007/annals.2003.158.977
  • [FZ05] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1-52. MR 2110627 (2005i:16065), https://doi.org/10.1215/S0012-7094-04-12611-9
  • [FZ07] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112-164. MR 2295199 (2008d:16049), https://doi.org/10.1112/S0010437X06002521
  • [Hum90] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
  • [Kel12] Bernhard Keller, Cluster algebras and derived categories, Derived categories in algebraic geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, pp. 123-183. MR 3050703
  • [KM04] Allen Knutson and Ezra Miller, Subword complexes in Coxeter groups, Adv. Math. 184 (2004), no. 1, 161-176. MR 2047852 (2005c:20066), https://doi.org/10.1016/S0001-8708(03)00142-7
  • [MRZ03] Robert Marsh, Markus Reineke, and Andrei Zelevinsky, Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4171-4186. MR 1990581 (2004g:52014), https://doi.org/10.1090/S0002-9947-03-03320-8
  • [PP12] Vincent Pilaud and Michel Pocchiola, Multitriangulations, pseudotriangulations and primitive sorting networks, Discrete Comput. Geom. 48 (2012), no. 1, 142-191. MR 2917206, https://doi.org/10.1007/s00454-012-9408-6
  • [PS11] Vincent Pilaud and Christian Stump.
    Brick polytopes of spherical subword complexes: A new approach to generalized associahedra.
    Preprint, arXiv:1111.3349 (v2), 2011.
  • [PS12] Vincent Pilaud and Francisco Santos, The brick polytope of a sorting network, European J. Combin. 33 (2012), no. 4, 632-662. MR 2864447, https://doi.org/10.1016/j.ejc.2011.12.003
  • [Rea07] Nathan Reading, Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5931-5958. MR 2336311 (2009d:20093), https://doi.org/10.1090/S0002-9947-07-04319-X

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 13F60, 20F55, 05E15, 05E45

Retrieve articles in all journals with MSC (2010): 13F60, 20F55, 05E15, 05E45


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: https://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

American Mathematical Society