The Characteristic Quasi-Polynomials of the Arrangements of Root Systems and Mid-Hyperplane Arrangements

  • Hidehiko Kamiya
  • Akimichi Takemura
  • Hiroaki Terao
Part of the Progress in Mathematics book series (PM, volume 283)


Let q be a positive integer. In [8], we proved that the cardinality of the complement of an integral arrangement, after the modulo q reduction, is a quasi-polynomial of q, which we call the characteristic quasi-polynomial. In this paper, we study general properties of the characteristic quasi-polynomial as well as discuss two important examples: the arrangements of reflecting hyperplanes arising from irreducible root systems and the mid-hyperplane arrangements. In the root system case, we present a beautiful formula for the generating function of the characteristic quasi-polynomial which has been essentially obtained by Ch. Athanasiadis [2] and by A. Blass and B. Sagan [3]. On the other hand, it is hard to find the generating function of the characteristic quasi-polynomial in the mid-hyperplane arrangement case. We determine them when the dimension is less than six.

Mathematics Subject Classification (2000)

32S22 05B35 17B20 


Characteristic quasi-polynomial elementary divisor hyperplane arrangement root system mid-hyperplane arrangement 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Aigner, Combinatorial Theory, Springer, Berlin, 1979.MATHGoogle Scholar
  2. [2]
    C. A. Athanasiadis, Characteristic polynomials of subspace arrangements and finite fields, Adv. Math. 122 (1996), 193–233.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Blass and B. Sagan, Characteristic and Ehrhart polynomials, J. Algebraic Combin. 7 (1998), 115–126.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    N. Bourbaki, Lie Groups and Lie Algebras: Chapters 4–6, Springer-Verlag, Berlin-Heidelberg-New York, 2002MATHGoogle Scholar
  5. [5]
    C. H. Coombs, A Theory of Data, John Wiley & Sons, New York, 1964.Google Scholar
  6. [6]
    M. Haiman, Conjectures on the quotient ring of diagonal invariants, J. Algebraic Combin. 3 (1994), 17–76.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Kamiya, P. Orlik, A. Takemura and H. Terao, Arrangements and ranking patterns, Ann. Comb. 10 (2006), 219–235.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    H. Kamiya, A. Takemura and H. Terao, Periodicity of hyperplane arrangements with integral coefficients modulo positive integers, J. Alg. Combin. 27 (2008), 317–330.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin, 1992.MATHGoogle Scholar
  10. [10]
    PARI/GP ( Scholar
  11. [11]
    R. Stanley, Enumerative Combinatorics, vol. I, Cambridge University Press, Cambridge, 1997.MATHGoogle Scholar
  12. [12]
    R. Suter, The number of lattice points in alcoves and the exponents of the finite Weyl group, Math. Comp., 67 (1998), 751–758.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Hidehiko Kamiya
    • 1
  • Akimichi Takemura
    • 2
  • Hiroaki Terao
    • 3
  1. 1.Faculty of EconomicsOkayama UniversityOkayamaJapan
  2. 2.Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  3. 3.Department of MathematicsHokkaido UniversitySapporoJapan

Personalised recommendations