Abstract
Let \(\mathcal {A}\) be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of \(\mathcal {A}\) are isometric. We prove that \(\mathcal {A}\) is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open question whether this suffices to characterize Coxeter arrangements in higher dimensions. We also present the three families of affine arrangements in the plane which are not reflection arrangements, but in which all the regions are isometric.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Björner, A., Brenti, F.: Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)
Björner, A., Las Vegas, M., Sturmfels, B., White, N., Ziegler, G.: Oriented Matroids, 2nd edn. Encyclopedia of Mathematics and its Applications, 46. Cambridge University Press, Cambridge (1999)
Davis, M.: The Geometry and Topology of Coxeter Groups, London Mathematical Society Monographs Series, 32. Princeton University Press, Princeton (2008)
Drton, M., Klivans, C.: A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Am. Math. Soc. 138(8), 2873–2887 (2010)
Grove, L.C., Benson, C.T.: Finite Reflection Groups, Graduate Texts in Mathematics, vol. 99. Springer, New York (1985)
Humphreys, J.: Reflection Groups and Coxeter Groups, Cambridge Studies in advanced mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
Klivans, C., Swartz, E.: Projection volumes of hyperplane arrangements. Discrete Comput. Geom. 46(3), 417–426 (2011)
Shannon, R.W.: Simplicial cells in arrangements of hyperplanes. Geom. Dedicata 8(2), 179–187 (1979)
West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall Inc, Upper Saddle River (2001)
Acknowledgments
Richard Ehrenborg was partially supported by National Security Agency Grant H98230-13-1-0280 and wishes to thank the Mathematics Department of Princeton University where he spent his sabbatical while this paper was written. Nathan Reading was supported by National Science Foundation Grant DMS-1101568.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ehrenborg, R., Klivans, C. & Reading, N. Coxeter arrangements in three dimensions. Beitr Algebra Geom 57, 891–897 (2016). https://doi.org/10.1007/s13366-016-0286-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-016-0286-6