Order

, Volume 19, Issue 1, pp 73–100

Order Dimension, Strong Bruhat Order and Lattice Properties for Posets

  • Nathan Reading
Article

Abstract

We determine the order dimension of the strong Bruhat order on finite Coxeter groups of types A, B and H. The order dimension is determined using a generalization of a theorem of Dilworth: dim (P)=width(Irr(P)), whenever P satisfies a simple order-theoretic condition called here the dissective property (or “clivage”). The result for dissective posets follows from an upper bound and lower bound on the dimension of any finite poset. The dissective property is related, via MacNeille completion, to the distributive property of lattices. We show a similar connection between quotients of the strong Bruhat order with respect to parabolic subgroups and lattice quotients.

Bruhat clivage congruence Coxeter critical complex Dilworth dissective distributive lattice MacNeille completion monotone triangle order dimension 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Billey, S., Jockusch, W. and Stanley, R. (1993) Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2(4), 345–374.Google Scholar
  2. 2.
    Birkhoff, G. (1973) Lattice Theory, 3rd edn, Amer. Math. Soc. Colloq. Publ. 25, Amer. Math. Soc.Google Scholar
  3. 3.
    Björner, A. and Brenti, F., Combinatorics of Coxeter Groups, Graduate Texts in Math., Springer-Verlag, to appear.Google Scholar
  4. 4.
    Björner, A. and Brenti, F. (1996) An improved tableau criterion for Bruhat order, Electron. J. Combin. 3(1), Research Paper 22.Google Scholar
  5. 5.
    Björner, A. and Wachs, M. (1988) Generalized quotients in Coxeter groups, Trans. Amer. Math. Soc. 308(1), 1–37.Google Scholar
  6. 6.
    Björner, A. and Wachs, M. (1997) Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc. 349(10), 3945–3975.Google Scholar
  7. 7.
    Chajda, I. and Snášel, V. (1998) Congruences in ordered sets, Math. Bohem. 123(1), 95–100.Google Scholar
  8. 8.
    Deodhar, R. and Srinivasan, M. (2001) A statistic on involutions, J. Algebraic Combin. 13(2), 187–198.Google Scholar
  9. 9.
    Dilworth, R. (1950) A decomposition theorem for partially ordered sets, Ann. of Math. (2) 51, 161–165.Google Scholar
  10. 10.
    Elkies, N., Kuperberg, G., Larsen, M. and Propp, J. (1992) Alternating sign matrices and domino tilings (Part I), J. Algebraic Combin. 1(2), 111–132.Google Scholar
  11. 11.
    Fan, K. (1972) On Dilworth's coding theorem, Math Z. 127, 92–94.Google Scholar
  12. 12.
    Felsner, S. and Trotter, W. (2000) Dimension, graph and hypergraph coloring, Order 17(2), 167–177.Google Scholar
  13. 13.
    Flath, S. (1993) The order dimension of multinomial lattices, Order 10(3), 201–219.Google Scholar
  14. 14.
    Garsia, A. and Stanton, D. (1984) Group actions on Stanley-Reisner rings and invariants of permutation groups, Adv. Math. 51(2), 107–201.Google Scholar
  15. 15.
    Geck, M., Hiss, G., Lübeck, F., Malle, G. and Pfeiffer, G. (1996) CHEVIE-A system for computing and processing generic character tables for finite groups of Lie type, in Weyl Groups and Hecke Algebras, AAECC 7, pp. 175–210.Google Scholar
  16. 16.
    Geck, M. and Kim, S. (1997) Bases for the Bruhat-Chevalley order on all finite Coxeter groups, J. Algebra 197(1), 278–310.Google Scholar
  17. 17.
    Grätzer, G. (1998) General Lattice Theory, 2nd edn, Birkhauser, Boston.Google Scholar
  18. 18.
    Hoffman, K. and Padberg, M. (2001) Set covering, packing and partitioning problems, in C. A. Floudas and M. Pardalos (eds), Encyclopedia of Optimization, Kluwer Academic Publishers, Dordrecht.Google Scholar
  19. 19.
    Humphreys, J. (1990) Reflection Groups and Coxeter Groups, Cambridge Stud. in Adv. Math. 29, Cambridge Univ. Press.Google Scholar
  20. 20.
    Kung, J., Personal communication.Google Scholar
  21. 21.
    Lascoux, A. and Schützenberger, M.-P. (1996) Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3, #R27.Google Scholar
  22. 22.
    MacNeille, H. (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42(3), 416–460.Google Scholar
  23. 23.
    Markowsky, G. (1992) Primes, irreducibles and extremal lattices, Order 9(3), 265–290.Google Scholar
  24. 24.
    Okada, S. (1993) Alternating sign matrices and some deformations of Weyl's denominator formulas, J. Algebraic Combin. 2(2), 155–176.Google Scholar
  25. 25.
    Rabinovitch, I. and Rival, I. (1979) The rank of a distributive lattice, Discrete Math. 25(3), 275–279.Google Scholar
  26. 26.
    Reading, N. (2002) Lattice and order properties of the poset of regions in a hyperplane arrangement, Preprint.Google Scholar
  27. 27.
    Reading, N. (2002) On the structure of Bruhat order, Ph.D. dissertation, University of Minnesota.Google Scholar
  28. 28.
    Robbins, D. (1991) The story of 1, 2, 7, 42, 429, 7436,..., Math. Intelligencer 13(2), 12–19.Google Scholar
  29. 29.
    Rozen, V. (1991) Coding of ordered sets, Ordered Sets and Lattices 10, 88–96 (Russian).Google Scholar
  30. 30.
    Schönert, M. et al. (1995) GAP-Groups, Algorithms, and Programming, 5th edn, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany.Google Scholar
  31. 31.
    Simion, R. (1999) A type-B associahedron, Preprint.Google Scholar
  32. 32.
    Stanley, R. (1997) Enumerative Combinatorics, Vol. I, Cambridge Stud. in Adv. Math. 49, Cambridge Univ. Press.Google Scholar
  33. 33.
    Trotter, W. (1992) Combinatorics and Partially Ordered Sets: Dimension Theory, Johns Hopkins Series in the Math. Sci., The Johns Hopkins Univ. Press.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Nathan Reading
    • 1
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisU.S.A.

Personalised recommendations