, 26:319 | Cite as

An Explicit Derivation of the Möbius Function for Bruhat Order

  • Brant C. Jones
Open Access


We give an explicit nonrecursive complete matching for the Hasse diagram of the strong Bruhat order of any interval in any Coxeter group. This yields a new derivation of the Möbius function, recovering a classical result due to Verma.


Bruhat order Matching 


  1. 1.
    Armstrong, D.: The sorting order on a Coxeter group. J. Comb. Theory Ser. A. (2009). doi: 10.1016/j.jcta.2009.03.009 Google Scholar
  2. 2.
    Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)Google Scholar
  3. 3.
    Björner, A.: Posets, regular CW complexes and Bruhat order. Eur. J. Comb. 5(1), 7–16 (1984)MATHGoogle Scholar
  4. 4.
    Björner, A., Wachs, M.: Bruhat order of Coxeter groups and shellability. Adv. Math. 43(1), 87–100 (1982)MATHCrossRefGoogle Scholar
  5. 5.
    Billey, S., Warrington, G.S.: Kazhdan–Lusztig polynomials for 321-hexagon-avoiding permutations. J. Algebr. Comb. 13(2), 111–136 (2001)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chari, M.K.: On discrete Morse functions and combinatorial decompositions. Discrete Math. 217(1–3), 101–113 (2000), Formal Power Series and Algebraic Combinatorics (Vienna, 1997)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Deodhar, V.V.: Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function. Invent. Math 39(2), 187–198 (1977)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Deodhar, V.V.: A combinatorial setting for questions in Kazhdan–Lusztig theory. Geom. Dedic. 36(1), 95–119 (1990)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dyer, M.J. Hecke algebras and shellings of Bruhat intervals. Compos. Math. 89(1), 91–115 (1993)MATHMathSciNetGoogle Scholar
  10. 10.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)Google Scholar
  11. 11.
    Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math 53(2), 165–184 (1979)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Marietti, M.: Algebraic and combinatorial properties of zircons. J. Algebr. Comb. 26(3), 363–382 (2007)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Marsh, R.J., Rietsch, K.: Parametrizations of flag varieties. Representat. Theory 8, 212–242 (2004) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Rietsch, K., Williams, L.: Discrete Morse theory for totally non-negative flag varieties. arXiv:0810.4314 [math.CO] (2008)
  15. 15.
    Stembridge, J.R.: A short derivation of the Möbius function for the Bruhat order. J. Algebr. Comb. 25(2), 141–148 (2007)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Verma, D.N.: Möbius inversion for the Bruhat ordering on a Weyl group. Ann. Sci. Éc. Norm. Super. (4) 4, 393–398 (1971)MATHGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of Mathematics, One Shields AvenueUniversity of CaliforniaDavisUSA

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