Advertisement

Order

, Volume 21, Issue 4, pp 315–344 | Cite as

Lattice Congruences of the Weak Order

  • Nathan ReadingEmail author
Article

Abstract

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 ηK: wwK be the projection onto the parabolic subgroup WK. We show that the fibers of ηK constitute the smallest lattice congruence with 1≡s for every s∈(SK). 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.

Keywords

Cambrian lattice congruence uniform Coxeter group parabolic subgroup poset of regions shard simplicial hyperplane arrangement Tamari lattice weak order 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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. Google Scholar
  2. 2.
    Björner, A., Edelman, P. and Ziegler, G.: Hyperplane arrangements with a lattice of regions, Discrete Comput. Geom. 5 (1990), 263–288. CrossRefGoogle Scholar
  3. 3.
    Björner, A. and Wachs, M.: Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc. 349(10) (1997), 3945–3975. CrossRefGoogle Scholar
  4. 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. CrossRefGoogle Scholar
  5. 5.
    Chajda, I. and Snášel, V.: Congruences in ordered sets, Math. Bohem. 123(1) (1998), 95–100. Google Scholar
  6. 6.
    Day, A.: Congruence normality: the characterization of the doubling class of convex sets, Algebra Universalis 31(3) (1994), 397–406. CrossRefGoogle Scholar
  7. 7.
    Edelman, P.: A partial order on the regions of ℝn dissected by hyperplanes, Trans. Amer. Math. Soc. 283(2) (1984), 617–631. Google Scholar
  8. 8.
    Fomin, S. and Zelevinsky, A.: Y-systems and generalized associahedra, Ann. of Math. (2) 158(3) (2003), 977–1018. Google Scholar
  9. 9.
    Freese, R., Ježek, J. and Nation, J.: Free Lattices, Math. Surveys Monographs 42, Amer. Math. Soc., 1995. Google Scholar
  10. 10.
    Funayama, N. and Nakayama, T.: On the distributivity of a lattice of lattice-congruences, Proc. Imp. Acad. Tokyo 18 (1942), 553–554. Google Scholar
  11. 11.
    Geyer, W.: On Tamari lattices, Discrete Math. 133(1–3) (1994), 99–122. CrossRefGoogle Scholar
  12. 12.
    Grätzer, G.: General Lattice Theory, 2nd edn, Birkhäuser Verlag, Basel, 1998. Google Scholar
  13. 13.
    Humphreys, J.: Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics 29, Cambridge Univ. Press, 1990. Google Scholar
  14. 14.
    Jedlička, P.: A combinatorial construction of the weak order of a Coxeter group, Preprint, 2003. Google Scholar
  15. 15.
    Malvenuto, C. and Reutenauer, C.: Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177(3) (1995), 967–982. CrossRefGoogle Scholar
  16. 16.
    Orlik, P. and Terao, H.: Arrangements of Hyperplanes, Grundlehren Math. Wiss. 300, Springer-Verlag, Berlin, 1992. Google Scholar
  17. 17.
    Reading, N.: Order dimension, strong Bruhat order and lattice properties for posets, Order 19(1) (2002), 73–100. CrossRefGoogle Scholar
  18. 18.
    Reading, N.: Lattice and order properties of the poset of regions in a hyperplane arrangement, Algebra Universalis 50(2) (2003), 179–205. CrossRefGoogle Scholar
  19. 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. CrossRefGoogle Scholar
  20. 20.
    Reading, N.: Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A 110(2) (2005), 237–273. CrossRefMathSciNetGoogle Scholar
  21. 21.
    Reading, N.: Cambrian lattices, Adv. Math., to appear. Google Scholar
  22. 22.
    Sloane, N. J. A. (ed.): The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/~njas/sequences/, 2003.
  23. 23.
    Stembridge, J.: A Maple package for posets, available electronically at http://www.math.lsa.umich.edu/~jrs/maple.html.
  24. 24.
    Thomas, H.: Tamari lattices for types B and D, Preprint math.CO/0311334, 2003. Google Scholar
  25. 25.
    Wilf, H.: The patterns of permutations, in: Kleitman and Combinatorics: A Celebration, Discrete Math. 257(2–3) (2002), 575–583. Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of MichiganAnn ArborUSA

Personalised recommendations