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The facial weak order and its lattice quotients. (English) Zbl 1375.05270
Summary: 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. Bj√∂rner [Contemp. Math. 34, 175–195 (1984; Zbl 0594.20029)] 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.

MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20F55 Reflection and Coxeter groups (group-theoretic aspects)
06A07 Combinatorics of partially ordered sets
05B05 Combinatorial aspects of block designs
03G10 Logical aspects of lattices and related structures
20F65 Geometric group theory
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[1] Abramenko, Peter; Brown, Kenneth S., Buildings, Graduate Texts in Mathematics 248, xxii+747 pp., (2008), Springer, New York · Zbl 1214.20033
[2] Bj\"orner, Anders; Brenti, Francesco, Combinatorics of Coxeter groups, Graduate Texts in Mathematics 231, xiv+363 pp., (2005), Springer, New York · Zbl 1110.05001
[3] Bj\`‘orner, Anders; Edelman, Paul H.; Ziegler, G\'’unter M., Hyperplane arrangements with a lattice of regions, Discrete Comput. Geom., 5, 3, 263-288, (1990) · Zbl 0698.51010
[4] Bj\"orner, Anders, Orderings of Coxeter groups. Combinatorics and algebra, Boulder, Colo., 1983, Contemp. Math. 34, 175-195, (1984), Amer. Math. Soc., Providence, RI
[5] Bourbaki, N., \'El\'ements de math\'ematique. Fasc. XXXIV. Groupes et alg\`ebres de Lie. Chapitre IV: Groupes de Coxeter et syst\`emes de Tits. Chapitre V: Groupes engendr\'es par des r\'eflexions. Chapitre VI: syst\`emes de racines, Actualit\'es Scientifiques et Industrielles, No. 1337, 288 pp. (loose errata) pp., (1968), Hermann, Paris · Zbl 0186.33001
[6] Chatel, Gr\'egory; Pilaud, Vincent, Cambrian Hopf algebras, Adv. Math., 311, 598-633, (2017) · Zbl 1369.05211
[7] Davis, Michael W., The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series 32, xvi+584 pp., (2008), Princeton University Press, Princeton, NJ · Zbl 1142.20020
[8] [Dye11]Dyer-WeakOrder Matthew Dyer, On the weak order of Coxeter groups, Preprint, <span class=”texttt”>a</span>rXiv:1108.5557, 2011.
[9] Freese, Ralph; Je\v zek, Jaroslav; Nation, J. B., Free lattices, Mathematical Surveys and Monographs 42, viii+293 pp., (1995), American Mathematical Society, Providence, RI · Zbl 0839.06005
[10] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15, 2, 497-529, (2002) · Zbl 1021.16017
[11] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. II. Finite type classification, Invent. Math., 154, 1, 63-121, (2003) · Zbl 1054.17024
[12] Geck, Meinolf; Pfeiffer, G\"otz, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series 21, xvi+446 pp., (2000), The Clarendon Press, Oxford University Press, New York · Zbl 0996.20004
[13] Hohlweg, Christophe; Labb\'e, Jean-Philippe, On inversion sets and the weak order in Coxeter groups, European J. Combin., 55, 1-19, (2016) · Zbl 1333.05143
[14] Hohlweg, Christophe; Lange, Carsten E. M. C.; Thomas, Hugh, Permutahedra and generalized associahedra, Adv. Math., 226, 1, 608-640, (2011) · Zbl 1233.20035
[15] Hohlweg, Christophe, Permutahedra and associahedra: generalized associahedra from the geometry of finite reflection groups. Associahedra, Tamari lattices and related structures, Prog. Math. Phys. 299, 129-159, (2012), Birkh\"auser/Springer, Basel · Zbl 1271.52012
[16] Humphreys, James E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, xii+204 pp., (1990), Cambridge University Press, Cambridge · Zbl 0725.20028
[17] [KLN01]KrobLatapyNovelliPhanSchwer 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.
[18] [NT06]NovelliThibon-trialgebras 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.
[19] Palacios, Patricia; Ronco, Mar\'\i a. O., Weak Bruhat order on the set of faces of the permutohedron and the associahedron, J. Algebra, 299, 2, 648-678, (2006) · Zbl 1110.16046
[20] Pilaud, Vincent; Stump, Christian, Brick polytopes of spherical subword complexes and generalized associahedra, Adv. Math., 276, 1-61, (2015) · Zbl 1405.05196
[21] Reading, Nathan, Lattice congruences of the weak order, Order, 21, 4, 315-344 (2005), (2004) · Zbl 1097.20036
[22] Reading, Nathan, Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A, 110, 2, 237-273, (2005) · Zbl 1133.20027
[23] Reading, Nathan, Cambrian lattices, Adv. Math., 205, 2, 313-353, (2006) · Zbl 1106.20033
[24] Reading, Nathan, Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc., 359, 12, 5931-5958, (2007) · Zbl 1189.05022
[25] Reading, Nathan, Sortable elements and Cambrian lattices, Algebra Universalis, 56, 3-4, 411-437, (2007) · Zbl 1184.20038
[26] Reading, Nathan, From the Tamari lattice to Cambrian lattices and beyond. Associahedra, Tamari lattices and related structures, Prog. Math. Phys. 299, 293-322, (2012), Birkh\"auser/Springer, Basel · Zbl 1292.20044
[27] Reading, Nathan; Speyer, David E., Cambrian fans, J. Eur. Math. Soc. (JEMS), 11, 2, 407-447, (2009) · Zbl 1213.20038
[28] Stanley, Richard P., Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics 49, xiv+626 pp., (2012), Cambridge University Press, Cambridge · Zbl 1247.05003
[29] Stella, Salvatore, Polyhedral models for generalized associahedra via Coxeter elements, J. Algebraic Combin., 38, 1, 121-158, (2013) · Zbl 1268.05242
[30] Ziegler, G\"unter M., Lectures on polytopes, Graduate Texts in Mathematics 152, x+370 pp., (1995), Springer-Verlag, New York · Zbl 0823.52002
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