Advertisement

Annals of Combinatorics

, 15:449 | Cite as

Periodicity of Non-Central Integral Arrangements Modulo Positive Integers

  • Hidehiko Kamiya
  • Akimichi Takemura
  • Hiroaki Terao
Article

Abstract

An integral coefficient matrix determines an integral arrangement of hyperplanes in \({\mathbb{R}^m}\) . After modulo q reduction \({(q \in {\mathbb{Z}_{ >0 }})}\) , the same matrix determines an arrangement \({\mathcal{A}_q}\) of “hyperplanes” in \({\mathbb{Z}^m_q}\) . In the special case of central arrangements, Kamiya, Takemura, and Terao [J. Algebraic Combin. 27(3), 317–330 (2008)] showed that the cardinality of the complement of \({\mathcal{A}_q}\) in \({\mathbb{Z}^m_q}\) is a quasi-polynomial in \({q \in {\mathbb{Z}_{ >0 }}}\) . Moreover, they proved in the central case that the intersection lattice of \({\mathcal{A}_q}\) is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement \({\hat{\mathcal{B}}_m^{[0,a]}}\) of Athanasiadis [J. Algebraic Combin. 10(3), 207–225 (1999)] to illustrate our results.

Mathematics Subject Classification

32S22 52C35 

Keywords

characteristic quasi-polynomial elementary divisor hyperplane arrangement intersection poset 

References

  1. 1.
    Athanasiadis C.A.: Extended Linial hyperplane arrangements for root systems and a conjecture of Postnikov and Stanley. J. Algebraic Combin. 10(3), 207–225 (1999)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Kamiya H., Orlik P., Takemura A., Terao H.: Arrangements and ranking patterns. Ann. Combin. 10(2), 219–235 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Kamiya H., Takemura A., Terao H.: Periodicity of hyperplane arrangements with integral coefficients modulo positive integers. J. Algebraic Combin. 27(3), 317–330 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Kamiya H., Takemura A., Terao H.: The characteristic quasi-polynomials of the arrangements of root systems and mid-hyperplane arrangements. In: El Zein, F., Suciu, A., Tosun, M., Uludaǧ, A.M., Yuzvinsky, S. (eds) Arrangements, Local Systems and Singularities., pp. 177–190. Birkhäuser, Basel (2010)Google Scholar
  5. 5.
    Orlik P., Terao H.: Arrangements of Hyperplanes. Springer-Verlag, Berlin (1992)MATHGoogle Scholar
  6. 6.
  7. 7.
    Stanley R.: Enumerative Combinatorics. Vol. I. Cambridge University Press, Cambridge (1997)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Hidehiko Kamiya
    • 1
  • Akimichi Takemura
    • 2
  • Hiroaki Terao
    • 3
  1. 1.Graduate School of EconomicsNagoya UniversityNagoyaJapan
  2. 2.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  3. 3.Department of MathematicsHokkaido UniversitySapporoJapan

Personalised recommendations