The facial weak order and its lattice quotients

Dermenjian, Aram; Hohlweg, Christophe; Pilaud, Vincent

doi:10.1090/tran/7307

The facial weak order and its lattice quotients
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by Aram Dermenjian, Christophe Hohlweg and Vincent Pilaud PDF

Trans. Amer. Math. Soc. 370 (2018), 1469-1507 Request permission

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

Peter Abramenko and Kenneth S. Brown, Buildings, Graduate Texts in Mathematics, vol. 248, Springer, New York, 2008. Theory and applications. MR 2439729, DOI 10.1007/978-0-387-78835-7
Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
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, DOI 10.1007/BF02187790
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, DOI 10.1090/conm/034/777701
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 [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Grégory Chatel and Vincent Pilaud, Cambrian Hopf algebras, Adv. Math. 311 (2017), 598–633. MR 3628225, DOI 10.1016/j.aim.2017.02.027
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
Matthew Dyer, On the weak order of Coxeter groups, Preprint, arXiv:1108.5557, 2011.
Ralph Freese, Jaroslav Ježek, and J. B. Nation, Free lattices, Mathematical Surveys and Monographs, vol. 42, American Mathematical Society, Providence, RI, 1995. MR 1319815, DOI 10.1090/surv/042
Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. MR 1887642, DOI 10.1090/S0894-0347-01-00385-X
Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121. MR 2004457, DOI 10.1007/s00222-003-0302-y
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
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, DOI 10.1016/j.ejc.2016.01.002
Christophe Hohlweg, Carsten E. M. C. Lange, and Hugh Thomas, Permutahedra and generalized associahedra, Adv. Math. 226 (2011), no. 1, 608–640. MR 2735770, DOI 10.1016/j.aim.2010.07.005
Christophe Hohlweg, Permutahedra and associahedra: generalized associahedra from the geometry of finite reflection groups, Associahedra, Tamari lattices and related structures, Progr. Math., vol. 299, Birkhäuser/Springer, Basel, 2012, pp. 129–159. MR 3221538, DOI 10.1007/978-3-0348-0405-9_{8}
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
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.
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.
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, DOI 10.1016/j.jalgebra.2005.09.042
Vincent Pilaud and Christian Stump, Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math. 276 (2015), 1–61. MR 3327085, DOI 10.1016/j.aim.2015.02.012
Nathan Reading, Lattice congruences of the weak order, Order 21 (2004), no. 4, 315–344 (2005). MR 2209128, DOI 10.1007/s11083-005-4803-8
Nathan Reading, Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A 110 (2005), no. 2, 237–273. MR 2142177, DOI 10.1016/j.jcta.2004.11.001
Nathan Reading, Cambrian lattices, Adv. Math. 205 (2006), no. 2, 313–353. MR 2258260, DOI 10.1016/j.aim.2005.07.010
Nathan Reading, Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5931–5958. MR 2336311, DOI 10.1090/S0002-9947-07-04319-X
Nathan Reading, Sortable elements and Cambrian lattices, Algebra Universalis 56 (2007), no. 3-4, 411–437. MR 2318219, DOI 10.1007/s00012-007-2009-1
Nathan Reading, From the Tamari lattice to Cambrian lattices and beyond, Associahedra, Tamari lattices and related structures, Progr. Math., vol. 299, Birkhäuser/Springer, Basel, 2012, pp. 293–322. MR 3221544, DOI 10.1007/978-3-0348-0405-9_{1}5
Nathan Reading and David E. Speyer, Cambrian fans, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 2, 407–447. MR 2486939, DOI 10.4171/JEMS/155
Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112
Salvatore Stella, Polyhedral models for generalized associahedra via Coxeter elements, J. Algebraic Combin. 38 (2013), no. 1, 121–158. MR 3070123, DOI 10.1007/s10801-012-0396-7
Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1