Arrangements stable under the Coxeter groups
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 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.
- C. A. Athanasiadis, “Algebraic Combinatorics of Graph Spectra, Subspace Arrangements and Tutte Polynomials”, Ph.D. thesis, MIT, 1996.Google Scholar
- N. Bourbaki, “Groupes et Algebres de Lie”, Chapitres 4, 5 et 6, Hermann, Paris, 1968.Google Scholar
- 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
- 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
- D. R. Mazur, Combinatorics: A Guided Tour, Mathematical Association of America, Washington, D.C., 2010.Google Scholar
- S. Ovchinnikov, “Graphs and Cubes”, Springer Science+Business Media, New York, NY, 2011.Google Scholar
- R. P. Stanley, “Enumerative Combinatorics”, Vol. 1, 2nd ed., Cambridge University Press, Cambridge, 2012.Google Scholar
- 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
- 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