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Vertex barycenter of generalized associahedra


Authors: Vincent Pilaud and Christian Stump
Journal: Proc. Amer. Math. Soc. 143 (2015), 2623-2636
MSC (2010): Primary 52B15; Secondary 13F60, 52B05
DOI: https://doi.org/10.1090/S0002-9939-2015-12357-X
Published electronically: February 11, 2015
MathSciNet review: 3326042
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that the vertex barycenter of generalized associahedra and permutahedra coincide for any finite Coxeter system.


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  • [CFZ02] Frédéric Chapoton, Sergey Fomin, and Andrei Zelevinsky, Polytopal realizations of generalized associahedra, Dedicated to Robert V. Moody. Canad. Math. Bull. 45 (2002), no. 4, 537-566. MR 1941227 (2003j:52014), https://doi.org/10.4153/CMB-2002-054-1
  • [CLS11] 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
  • [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
  • [HL07] Christophe Hohlweg and Carsten E. M. C. Lange, Realizations of the associahedron and cyclohedron, Discrete Comput. Geom. 37 (2007), no. 4, 517-543. MR 2321739 (2008g:52021), https://doi.org/10.1007/s00454-007-1319-6
  • [HLR10] Christophe Hohlweg, Jonathan Lortie, and Annie Raymond, The centers of gravity of the associahedron and of the permutahedron are the same, Electron. J. Combin. 17 (2010), no. 1, Research Paper 72, 14. MR 2651725 (2011f:05334)
  • [HLT11] Christophe Hohlweg, Carsten E. M. C. Lange, and Hugh Thomas, Permutahedra and generalized associahedra, Adv. Math. 226 (2011), no. 1, 608-640. MR 2735770 (2012d:20085), https://doi.org/10.1016/j.aim.2010.07.005
  • [Hoh12] Christophe Hohlweg, Permutahedra and associahedra: generalized associahedra from the geometry of finite reflection groups, Associahedra, Tamari lattices and related structures, Prog. Math. Phys., vol. 299, Birkhäuser/Springer, Basel, 2012, pp. 129-159. MR 3221538, https://doi.org/10.1007/978-3-0348-0405-9_8
  • [Hum78] James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York, 1978. Second printing, revised. MR 499562 (81b:17007)
  • [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)
  • [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
  • [Lod04] Jean-Louis Loday, Realization of the Stasheff polytope, Arch. Math. (Basel) 83 (2004), no. 3, 267-278. MR 2108555 (2005g:52028), https://doi.org/10.1007/s00013-004-1026-y
  • [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 and generalized associahedra.
    To appear in Adv. Math. Preprint, arXiv:1111.3349, 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
  • [PS13] Vincent Pilaud and Christian Stump, EL-labelings and canonical spanning trees for subword complexes, Discrete geometry and optimization, Fields Inst. Commun., vol. 69, Springer, New York, 2013, pp. 213-248. MR 3156785, https://doi.org/10.1007/978-3-319-00200-2_13
  • [Rea06] Nathan Reading, Cambrian lattices, Adv. Math. 205 (2006), no. 2, 313-353. MR 2258260 (2007g:05195), https://doi.org/10.1016/j.aim.2005.07.010
  • [Rea07] Nathan Reading, Sortable elements and Cambrian lattices, Algebra Universalis 56 (2007), no. 3-4, 411-437. MR 2318219 (2008d:20073), https://doi.org/10.1007/s00012-007-2009-1
  • [RS09] Nathan Reading and David E. Speyer, Cambrian fans, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 2, 407-447. MR 2486939 (2011a:20102), https://doi.org/10.4171/JEMS/155
  • [Ste13] Salvatore Stella, Polyhedral models for generalized associahedra via Coxeter elements, J. Algebraic Combin. 38 (2013), no. 1, 121-158. MR 3070123, https://doi.org/10.1007/s10801-012-0396-7
  • [Stu11] Christian Stump, A new perspective on $ k$-triangulations, J. Combin. Theory Ser. A 118 (2011), no. 6, 1794-1800. MR 2793610 (2012e:52042), https://doi.org/10.1016/j.jcta.2011.03.001
  • [Woo04] A. Woo, Catalan numbers and Schubert polynomials for $ w=1(n+1)... 2$.
    Preprint, arXiv:0407160, 2004.

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Additional Information

Vincent Pilaud
Affiliation: CNRS and LIX, École Polytechnique, 91128 Palaiseau, France
Email: vincent.pilaud@lix.polytechnique.fr

Christian Stump
Affiliation: Institut für Algebra, Zahlentheorie, Diskrete Mathematik, Universität Hannover, Hannover, Germany
Address at time of publication: Diskrete Geometrie, Freie Universität Berlin, Arnimallee 2, 14195 Berlin, Germany
Email: stump@math.uni-hannover.de, christian.stump@fu-berlin.de

DOI: https://doi.org/10.1090/S0002-9939-2015-12357-X
Received by editor(s): October 16, 2012
Received by editor(s) in revised form: September 9, 2013
Published electronically: February 11, 2015
Additional Notes: The first 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).
Communicated by: Jim Haglund
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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