Lattice congruences of the weak order. (English) Zbl 1097.20036

Author’s summary: We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system \((W,S)\) and a subset \(K\subseteq S\), let \(\eta_K\colon w\mapsto w_K\) be the projection onto the parabolic subgroup \(W_K\). We show that the fibers of \(\eta_K\) constitute the smallest lattice congruence with \(1\equiv s\) for every \(s\in(S-K)\). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type \(A\) or \(B\) we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order.


20F55 Reflection and Coxeter groups (group-theoretic aspects)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
06A07 Combinatorics of partially ordered sets
08A30 Subalgebras, congruence relations


posets; OEIS
Full Text: DOI arXiv


[1] Björner, A.: Orderings of Coxeter groups, in: Combinatorics and Algebra (Boulder, Colo., 1983), Contemp. Math. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 175–195.
[2] Björner, A., Edelman, P. and Ziegler, G.: Hyperplane arrangements with a lattice of regions, Discrete Comput. Geom. 5 (1990), 263–288. · Zbl 0698.51010
[3] Björner, A. and Wachs, M.: Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc. 349(10) (1997), 3945–3975. · Zbl 0886.05126
[4] Caspard, N., Le Conte de Poly-Barbut, C. and Morvan, M.: Cayley lattices of finite Coxeter groups are bounded, Adv. in Appl. Math. 33(1) (2004), 71–94. · Zbl 1097.06001
[5] Chajda, I. and Snášel, V.: Congruences in ordered sets, Math. Bohem. 123(1) (1998), 95–100. · Zbl 0897.06004
[6] Day, A.: Congruence normality: the characterization of the doubling class of convex sets, Algebra Universalis 31(3) (1994), 397–406. · Zbl 0804.06006
[7] Edelman, P.: A partial order on the regions of \(\mathbb{R}\)n dissected by hyperplanes, Trans. Amer. Math. Soc. 283(2) (1984), 617–631. · Zbl 0555.06003
[8] Fomin, S. and Zelevinsky, A.: Y-systems and generalized associahedra, Ann. of Math. (2) 158(3) (2003), 977–1018. · Zbl 1057.52003
[9] Freese, R., Ježek, J. and Nation, J.: Free Lattices, Math. Surveys Monographs 42, Amer. Math. Soc., 1995.
[10] Funayama, N. and Nakayama, T.: On the distributivity of a lattice of lattice-congruences, Proc. Imp. Acad. Tokyo 18 (1942), 553–554. · Zbl 0063.01483
[11] Geyer, W.: On Tamari lattices, Discrete Math. 133(1–3) (1994), 99–122. · Zbl 0811.06005
[12] Grätzer, G.: General Lattice Theory, 2nd edn, Birkhäuser Verlag, Basel, 1998. · Zbl 0909.06002
[13] Humphreys, J.: Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics 29, Cambridge Univ. Press, 1990. · Zbl 0725.20028
[14] Jedlička, P.: A combinatorial construction of the weak order of a Coxeter group, Preprint, 2003.
[15] Malvenuto, C. and Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177(3) (1995), 967–982. · Zbl 0838.05100
[16] Orlik, P. and Terao, H.: Arrangements of Hyperplanes, Grundlehren Math. Wiss. 300, Springer-Verlag, Berlin, 1992. · Zbl 0757.55001
[17] Reading, N.: Order dimension, strong Bruhat order and lattice properties for posets, Order 19(1) (2002), 73–100. · Zbl 1007.05097
[18] Reading, N.: Lattice and order properties of the poset of regions in a hyperplane arrangement, Algebra Universalis 50(2) (2003), 179–205. · Zbl 1092.06006
[19] Reading, N.: The order dimension of the poset of regions in a hyperplane arrangement, J. Combin. Theory Ser. A 104(2) (2003), 265–285. · Zbl 1044.52010
[20] Reading, N.: Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A 110(2) (2005), 237–273. · Zbl 1133.20027
[21] Reading, N.: Cambrian lattices, Adv. Math., to appear. · Zbl 1106.20033
[22] Sloane, N. J. A. (ed.): The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/\(\sim\)njas/sequences/, 2003. · Zbl 1044.11108
[23] Stembridge, J.: A Maple package for posets, available electronically at http://www.math.lsa.umich.edu/\(\sim\)jrs/maple.html. · Zbl 0849.68068
[24] Thomas, H.: Tamari lattices for types B and D, Preprint math.CO/0311334, 2003.
[25] Wilf, H.: The patterns of permutations, in: Kleitman and Combinatorics: A Celebration, Discrete Math. 257(2–3) (2002), 575–583. · Zbl 1028.05002
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