Arrangements stable under the Coxeter groups

  • Hidehiko Kamiya
  • Akimichi Takemura
  • Hiroaki Terao
Part of the CRM Series book series (PSNS)


Let B be a real hyperplane arrangement which is stable under the action of a Coxeter group W. Then W acts naturally on the set of chambers of B. We assume that B is disjoint from the Coxeter arrangement A = A(W) of W. In this paper, we show that the W-orbits of the set of chambers of B are in one-to-one correspondence with the chambers of C = BB which are contained in an arbitrarily fixed chamber of A. From this fact, we find that the number of W-orbits of the set of chambers of B is given by the number of chambers of C divided by the order of W. We will also study the set of chambers of C which are contained in a chamber b of B. We prove that the cardinality of this set is equal to the order of the isotropy subgroup Wb of b. We illustrate these results with some examples, and solve an open problem in [H. Kamiya, A. Takemura, H. Terao, Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. 47 (2011) 379–400] by using our results.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. A. Athanasiadis, “Algebraic Combinatorics of Graph Spectra, Subspace Arrangements and Tutte Polynomials”, Ph.D. thesis, MIT, 1996.Google Scholar
  2. [2]
    C. A. Athanasiadis, Characteristic polynomials of subspace arrangements and finite fields, Adv. Math. 122 (1996) 193–233.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    N. Bourbaki, “Groupes et Algebres de Lie”, Chapitres 4, 5 et 6, Hermann, Paris, 1968.Google Scholar
  4. [4]
    J. L. Chandon, J. Lemaire and J. Pouget, Dénombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines 62 (1978), 61–80.MATHMathSciNetGoogle Scholar
  5. [5]
    H. Crapo and G.-C. Rota, “On the Foundations of Combinatorial Theory: Combinatorial Geometries”, preliminary edition, MIT Press, Cambridge, MA, 1970.MATHGoogle Scholar
  6. [6]
    R. A. Dean and G. Keller, Natural partial orders, Canad. J. Math. 20 (1968), 535–554.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    H. Kamiya, P. Orlik, A. Takemura and H. Terao, Arrangements and ranking patterns, Ann. Comb. 10 (2006), 219–235.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    H. Kamiya, A. Takemura and H. Terao, Periodicity of hyperplane arrangements with integral coefficients modulo positive integers, J. Algebraic Combin. 27 (2008) 317–330.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    H. Kamiya, A. Takemura and H. Terao, The characteristic quasi-polynomials of the arrangements of root systems and mid-hyperplane arrangements, In: F. El Zein, A. I. Suciu, M. Tosun, A. M. Uludağ, S. Yuzvinsky, (Eds.), “Arrangements, Local Systems and Singularities”, CIMPA Summer School, Galatasaray University, Istanbul, 2007, Progress in Mathematics 283, Birkhäuser Verlag, Basel, 2009, pp. 177–190.Google Scholar
  10. [10]
    H. Kamiya, A. Takemura and H. Terao, Periodicity of non-central integral arrangements modulo positive integers, Ann. Comb. 15 (2011), 449–464.CrossRefMATHMathSciNetGoogle Scholar
  11. [11]
    H. Kamiya, A. Takemura and H. Terao, Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. 47 (2011), 379–400.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    H. Kamiya, A. Takemura and N. Tokushige, Application of arrangement theory to unfolding models, In: “Arrangements of Hyperplanes (Sapporo 2009)”, H. Terao, S. Yuzvinsky (eds.), Proceedings of the Second MSJ-SI, Hokkaido University, Sapporo, 2009, Advanced Studies in Pure Math., 62, Math. Soc. of Japan, Tokyo, 2012, 399–415.Google Scholar
  13. [13]
    R. D. Luce, Semiorders and a theory of utility discrimination, Econometrica 24 (1956), 178–191.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    D. R. Mazur, Combinatorics: A Guided Tour, Mathematical Association of America, Washington, D.C., 2010.Google Scholar
  15. [15]
    P. Orlik and H. Terao, “Arrangements of Hyperplanes”, Springer-Verlag, Berlin, 1992.CrossRefGoogle Scholar
  16. [16]
    S. Ovchinnikov, Hyperplane arrangements in preference modeling, J. Math. Psych. 49 (2005), 481–488.CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    S. Ovchinnikov, “Graphs and Cubes”, Springer Science+Business Media, New York, NY, 2011.Google Scholar
  18. [18]
    A. Postnikov and R. P. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory Ser. A 91 (2000), 544–597.CrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    D. Scott and P. Suppes, Foundational aspects of theories of measurement, J. Symbolic Logic 23 (1958), 113–128.CrossRefMathSciNetGoogle Scholar
  20. [20]
    R. P. Stanley, “Enumerative Combinatorics”, Vol. 1, 2nd ed., Cambridge University Press, Cambridge, 2012.Google Scholar
  21. [21]
    R. P. Stanley, “Enumerative Combinatorics”, Vol. 2, Cambridge University Press, Cambridge, 1999.CrossRefGoogle Scholar
  22. [22]
    R. P. Stanley, An introduction to hyperplane arrangements, In: E. Miller, V. Reiner, B. Sturmfels, (Eds.), “Geometric Combinatorics, IAS/Park City Mathematics Series 13”, American Mathematical Society, Providence, RI, 2007, pp. 389–496.Google Scholar
  23. [23]
    R. L. Wine and J. E. Freund, On the enumeration of decision patterns involving n means, Ann. Math. Statist. 28 (1957), 256–259.CrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    T. Zaslavsky, Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1, no. 154 (1975).Google Scholar

Copyright information

© Scuola Normale Superiore Pisa 2012

Authors and Affiliations

  • Hidehiko Kamiya
  • Akimichi Takemura
  • Hiroaki Terao

There are no affiliations available

Personalised recommendations