Abstract
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 = B ∪ B 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 W b 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.
This work was partially supported by JSPS KAKENHI (22540134).
This research was supported by JST CREST.
This work was partially supported by JSPS KAKENHI (21340001).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. A. Athanasiadis, “Algebraic Combinatorics of Graph Spectra, Subspace Arrangements and Tutte Polynomials”, Ph.D. thesis, MIT, 1996.
C. A. Athanasiadis, Characteristic polynomials of subspace arrangements and finite fields, Adv. Math. 122 (1996) 193–233.
N. Bourbaki, “Groupes et Algebres de Lie”, Chapitres 4, 5 et 6, Hermann, Paris, 1968.
J. L. Chandon, J. Lemaire and J. Pouget, Dénombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines 62 (1978), 61–80.
H. Crapo and G.-C. Rota, “On the Foundations of Combinatorial Theory: Combinatorial Geometries”, preliminary edition, MIT Press, Cambridge, MA, 1970.
R. A. Dean and G. Keller, Natural partial orders, Canad. J. Math. 20 (1968), 535–554.
H. Kamiya, P. Orlik, A. Takemura and H. Terao, Arrangements and ranking patterns, Ann. Comb. 10 (2006), 219–235.
H. Kamiya, A. Takemura and H. Terao, Periodicity of hyperplane arrangements with integral coefficients modulo positive integers, J. Algebraic Combin. 27 (2008) 317–330.
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.
H. Kamiya, A. Takemura and H. Terao, Periodicity of non-central integral arrangements modulo positive integers, Ann. Comb. 15 (2011), 449–464.
H. Kamiya, A. Takemura and H. Terao, Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. 47 (2011), 379–400.
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.
R. D. Luce, Semiorders and a theory of utility discrimination, Econometrica 24 (1956), 178–191.
D. R. Mazur, Combinatorics: A Guided Tour, Mathematical Association of America, Washington, D.C., 2010.
P. Orlik and H. Terao, “Arrangements of Hyperplanes”, Springer-Verlag, Berlin, 1992.
S. Ovchinnikov, Hyperplane arrangements in preference modeling, J. Math. Psych. 49 (2005), 481–488.
S. Ovchinnikov, “Graphs and Cubes”, Springer Science+Business Media, New York, NY, 2011.
A. Postnikov and R. P. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory Ser. A 91 (2000), 544–597.
D. Scott and P. Suppes, Foundational aspects of theories of measurement, J. Symbolic Logic 23 (1958), 113–128.
R. P. Stanley, “Enumerative Combinatorics”, Vol. 1, 2nd ed., Cambridge University Press, Cambridge, 2012.
R. P. Stanley, “Enumerative Combinatorics”, Vol. 2, Cambridge University Press, Cambridge, 1999.
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.
R. L. Wine and J. E. Freund, On the enumeration of decision patterns involving n means, Ann. Math. Statist. 28 (1957), 256–259.
T. Zaslavsky, Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1, no. 154 (1975).
Editor information
Rights and permissions
Copyright information
© 2012 Scuola Normale Superiore Pisa
About this paper
Cite this paper
Kamiya, H., Takemura, A., Terao, H. (2012). Arrangements stable under the Coxeter groups. In: Bjorner, A., Cohen, F., De Concini, C., Procesi, C., Salvetti, M. (eds) Configuration Spaces. CRM Series. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-431-1_15
Download citation
DOI: https://doi.org/10.1007/978-88-7642-431-1_15
Publisher Name: Edizioni della Normale, Pisa
Print ISBN: 978-88-7642-430-4
Online ISBN: 978-88-7642-431-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)