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)

Abstract

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 

Keywords

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

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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

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