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)

 
 

 

The facial weak order and its lattice quotients


Authors: Aram Dermenjian, Christophe Hohlweg and Vincent Pilaud
Journal: Trans. Amer. Math. Soc. 370 (2018), 1469-1507
MSC (2010): Primary 05E99, 20F55; Secondary 06B99, 03G10
DOI: https://doi.org/10.1090/tran/7307
Published electronically: October 24, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the facial weak order, a poset structure that extends the weak order on a finite Coxeter group $ W$ to the set of all faces of the permutahedron of $ W$. We first provide three characterizations of this poset: the original one in terms of cover relations, the geometric one that generalizes the notion of inversion sets, and the combinatorial one as an induced subposet of the poset of intervals of the weak order. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Björner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of their classes. As application, we describe the facial boolean lattice on the faces of the cube and the facial Cambrian lattice on the faces of the corresponding generalized associahedron.


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

  • [AB08] Peter Abramenko and Kenneth S. Brown, Buildings: Theory and applications, Graduate Texts in Mathematics, vol. 248, Springer, New York, 2008. MR 2439729
  • [BB05] Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
  • [BEZ90] Anders Björner, Paul H. Edelman, and Günter M. Ziegler, Hyperplane arrangements with a lattice of regions, Discrete Comput. Geom. 5 (1990), no. 3, 263-288. MR 1036875, https://doi.org/10.1007/BF02187790
  • [Bjö84] Anders Björner, Orderings of Coxeter groups, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 175-195. MR 777701, https://doi.org/10.1090/conm/034/777701
  • [Bou68] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968. MR 0240238
  • [CP17] Grégory Chatel and Vincent Pilaud, Cambrian Hopf algebras, Adv. Math. 311 (2017), 598-633. MR 3628225, https://doi.org/10.1016/j.aim.2017.02.027
  • [Dav08] Michael W. Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008. MR 2360474
  • [Dye11] Matthew Dyer, On the weak order of Coxeter groups, Preprint, arXiv:1108.5557, 2011.
  • [FJN95] Ralph Freese, Jaroslav Ježek, and J. B. Nation, Free lattices, Mathematical Surveys and Monographs, vol. 42, American Mathematical Society, Providence, RI, 1995. MR 1319815
  • [FZ02] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529. MR 1887642, https://doi.org/10.1090/S0894-0347-01-00385-X
  • [FZ03] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63-121. MR 2004457, https://doi.org/10.1007/s00222-003-0302-y
  • [GP00] Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
  • [HL16] Christophe Hohlweg and Jean-Philippe Labbé, On inversion sets and the weak order in Coxeter groups, European J. Combin. 55 (2016), 1-19. MR 3474789, https://doi.org/10.1016/j.ejc.2016.01.002
  • [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, 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
  • [Hum90] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460
  • [KLN01] Daniel Krob, Matthieu Latapy, Jean-Christophe Novelli, Ha-Duong Phan, and Sylviane Schwer, Pseudo-Permutations I: First Combinatorial and Lattice Properties, 13th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2001), 2001.
  • [NT06] Jean-Christophe Novelli and Jean-Yves Thibon, Polynomial realizations of some trialgebras, 18th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2006), 2006.
  • [PR06] Patricia Palacios and María O. Ronco, Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra 299 (2006), no. 2, 648-678. MR 2228332, https://doi.org/10.1016/j.jalgebra.2005.09.042
  • [PS15] Vincent Pilaud and Christian Stump, Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math. 276 (2015), 1-61. MR 3327085, https://doi.org/10.1016/j.aim.2015.02.012
  • [Rea04] Nathan Reading, Lattice congruences of the weak order, Order 21 (2004), no. 4, 315-344 (2005). MR 2209128, https://doi.org/10.1007/s11083-005-4803-8
  • [Rea05] Nathan Reading, Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A 110 (2005), no. 2, 237-273. MR 2142177, https://doi.org/10.1016/j.jcta.2004.11.001
  • [Rea06] Nathan Reading, Cambrian lattices, Adv. Math. 205 (2006), no. 2, 313-353. MR 2258260, https://doi.org/10.1016/j.aim.2005.07.010
  • [Rea07a] Nathan Reading, Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5931-5958. MR 2336311, https://doi.org/10.1090/S0002-9947-07-04319-X
  • [Rea07b] Nathan Reading, Sortable elements and Cambrian lattices, Algebra Universalis 56 (2007), no. 3-4, 411-437. MR 2318219, https://doi.org/10.1007/s00012-007-2009-1
  • [Rea12] Nathan Reading, From the Tamari lattice to Cambrian lattices and beyond, Associahedra, Tamari lattices and related structures, Prog. Math. Phys., vol. 299, Birkhäuser/Springer, Basel, 2012, pp. 293-322. MR 3221544, https://doi.org/10.1007/978-3-0348-0405-9_15
  • [RS09] Nathan Reading and David E. Speyer, Cambrian fans, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 2, 407-447. MR 2486939, https://doi.org/10.4171/JEMS/155
  • [Sta12] Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112
  • [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
  • [Zie95] Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 05E99, 20F55, 06B99, 03G10

Retrieve articles in all journals with MSC (2010): 05E99, 20F55, 06B99, 03G10


Additional Information

Aram Dermenjian
Affiliation: LIX, École Polytechnique, Palaiseau & LaCIM, Université du Québec À Montréal (UQAM), Montréal, Québec H2H 2A9, Canada
Email: aram.dermenjian@gmail.com

Christophe Hohlweg
Affiliation: LaCIM, Université du Québec À Montréal (UQAM), Montréal, Québec H2H 2A9, Canada
Email: hohlweg.christophe@uqam.ca

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

DOI: https://doi.org/10.1090/tran/7307
Keywords: Permutahedra, weak order, Coxeter complex, lattice quotients, associahedra, Cambrian lattices.
Received by editor(s): February 16, 2016
Received by editor(s) in revised form: May 18, 2017
Published electronically: October 24, 2017
Additional Notes: The first author was partially supported by the French ANR grant EGOS (12JS0200201) and an ISM Graduate Scholarship. The second author was supported by NSERC Discovery grant Coxeter groups and related structures. The third author was partially supported by the French ANR grants EGOS (12JS0200201) and SC3A (15CE40000401).
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society