Abstract
We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present some series of arrangements related to the known arrangements in characteristic zero. We further enumerate simplicial arrangements with given symmetry groups. Finally, we determine all finite complex reflection groups affording combinatorially simplicial arrangements. It turns out that combinatorial simpliciality coincides with inductive freeness for finite complex reflection groups except for the Shephard–Todd group \(G_{31}\).
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Notes
Minimal fields of definition were computed in Cuntz (2011) for all known simplicial arrangements. Most of the occurring fields are of the form \(\mathbb {Q}(\zeta )\cap \mathbb {R}\) for \(\zeta \) a root of unity.
We use the word “incidence” instead of “matroid” in dimension three because we are then only interested in the relation between (projective) points and lines.
In \(K^3\) this is equivalent to the fact that all hyperplanes except one meet in one one-dimensional intersection point; see Orlik and Terao (1992), Definition 2.15].
References
Athanasiadis, C.A.: Characteristic polynomials of subspace arrangements and finite fields. Adv. Math. 122(2), 193–233 (1996)
Barakat, M., Cuntz, M.: Coxeter and crystallographic arrangements are inductively free. Adv. Math. 229(1), 691–709 (2012)
Cuntz, M.: Crystallographic arrangements: Weyl groupoids and simplicial arrangements. Bull. London Math. Soc. 43(4), 734–744 (2011)
Cuntz, M.: Minimal fields of definition for simplicial arrangements in the real projective plane. Innov. Incid. Geom. 12, 49–60 (2011)
Cuntz, M.: Simplicial arrangements with up to 27 lines. Discrete Comput. Geom. 48(3), 682–701 (2012)
Cuntz, M., Heckenberger, I.: Finite Weyl groupoids. J. Reine Angew. Math. (2010), p. 35., available at arXiv:1008.5291v1
Cuntz, M., Hoge, T.: Free but not recursively free arrangement. Proc. Am. Math. Soc. p. 7, available at arXiv:1302.4055 (2013)
Deligne, P.: Les immeubles des groupes de tresses généralisés. Invent. Math. 17, 273–302 (1972)
Grünbaum, B.: Arrangements of hyperplanes, Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1971) (Baton Rouge, La.), Louisiana State Univ., pp. 41–106 (1971)
Grünbaum, B.: A catalogue of simplicial arrangements in the real projective plane. Ars Math. Contemp. 2(1), 25 (2009)
Grünbaum, B.: Simplicial arrangements revisited, preprint, p. 10 (2013)
Hoge, T., Röhrle, G.: On inductively free reflection arrangements. J. Reine Angew. Math., p. 16, available at arXiv:1208.3131 (2012)
Karpilovsky, G.: The Schur multiplier. In: London Mathematical Society Monographs. New Series, vol. 2, The Clarendon Press, Oxford University Press, New York (1987)
Melchior, E.: Über Vielseite der projektiven Ebene. Deutsche Math. 5, 461–475 (1941)
Orlik, P., Terao, H.: Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, vol. 300. Springer, Berlin (1992)
Stanley, R.P.: An introduction to hyperplane arrangements. Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, pp. 389–496. MR: 2383131 (2007)
Steinberg, R.: The representations of GL(3, q), GL(4, q), PGL(3, q), and PGL(4, q). Can. J. Math. 3, 225–235 (1951)
Acknowledgments
We would like to thank U. Dempwolff and G. Malle for many valuable comments. The definition of combinatorial simpliciality in dimension greater than three was suggested by M. Falk.
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Appendix A: Some simplicial arrangements over \(\mathbb {C}\)
Appendix A: Some simplicial arrangements over \(\mathbb {C}\)
We reproduce here the newly found simplicial arrangements over \(\mathbb {C}\) in the most compact notation, namely as sets of normal vectors in the original finite field. The ordering is the same as in 5.2. The symbol \(\omega \) stands for a primitive element of the corresponding field. As in 5.2, we omit those incidences which come from real simplicial arrangements, from finite complex reflection groups, or which are near pencil.
\(q=5\), \(\,\, \{(0,1,1)\), \((1,0,0)\), \((1,0,1)\), \((1,0,2)\), \((1,1,0)\), \((1,2,1)\), \((1,3,2)\), \((1,4,0)\), \((1,4,1)\), \((1,4,2)\), \((1,4,3)\), \((1,4,4)\}\),
\(q=7\), \(\,\, \{(0,1,0)\), \((0,1,2)\), \((0,1,3)\), \((0,1,4)\), \((1,0,0)\), \((1,0,1)\), \((1,1,3)\), \((1,2,0)\), \((1,2,2)\), \((1,3,0)\), \((1,4,1)\), \((1,5,1)\), \((1,6,3)\), \((1,6,5)\}\),
\(q=7\), \(\,\, \{(0,1,0)\), \((0,1,2)\), \((0,1,3)\), \((0,1,4)\), \((1,0,0)\), \((1,0,1)\), \((1,0,5)\), \((1,1,3)\), \((1,2,0)\), \((1,2,2)\), \((1,3,0)\), \((1,4,1)\), \((1,5,1)\), \((1,6,3)\), \((1,6,5)\}\),
\(q=7\), \(\,\, \{(0,0,1)\), \((1,0,0)\), \((1,0,5)\), \((1,0,6)\), \((1,1,0)\), \((1,1,2)\), \((1,1,6)\), \((1,2,0)\), \((1,2,2)\), \((1,2,6)\), \((1,3,0)\), \((1,3,5)\), \((1,5,1)\), \((1,5,3)\), \((1,5,4)\}\),
\(q=7\), \(\,\, \{(0,1,2)\), \((1,0,0)\), \((1,0,2)\), \((1,1,4)\), \((1,1,5)\), \((1,1,6)\), \((1,2,3)\), \((1,3,0)\), \((1,3,2)\), \((1,3,4)\), \((1,4,0)\), \((1,4,1)\), \((1,4,4)\), \((1,5,6)\), \((1,6,6)\}\),
\(q = 9\), \(\,\, \{(0,0,1)\), \((1,0,0)\), \((1,0,\omega ^2)\), \((1,0,2)\), \((1,1,2)\), \((1,\omega ^3,2)\), \((1,\omega ^3,\omega ^6)\), \((1,2,\omega ^3)\), \((1,2,\omega ^5)\), \((1,\omega ^5,0)\), \((1,\omega ^5,1)\), \((1,\omega ^5,2)\), \((1,\omega ^6,0)\),
\((1,\omega ^6,\omega ^5)\), \((1,\omega ^6,\omega ^7)\}\),
\(q=7\), \(\,\, \{(0,1,2)\), \((1,0,0)\), \((1,1,4)\), \((1,1,5)\), \((1,1,6)\), \((1,2,2)\), \((1,2,3)\), \((1,3,0)\), \((1,3,4)\), \((1,4,0)\), \((1,4,1)\), \((1,4,4)\), \((1,5,6)\), \((1,6,2)\), \((1,6,6)\}\),
\(q=7\), \(\,\, \{(0,0,1)\), \((0,1,2)\), \((1,0,1)\), \((1,2,0)\), \((1,2,2)\), \((1,2,6)\), \((1,3,0)\), \((1,4,0)\), \((1,4,1)\), \((1,4,2)\), \((1,5,0)\), \((1,5,2)\), \((1,5,3)\), \((1,6,2)\), \((1,6,4)\), \((1,6,5)\}\),
\(q=7\), \(\,\, \{(0,0,1)\), \((0,1,2)\), \((1,0,0)\), \((1,0,1)\), \((1,2,0)\), \((1,2,2)\), \((1,2,6)\), \((1,3,0)\), \((1,4,0)\), \((1,4,1)\), \((1,4,2)\), \((1,5,0)\), \((1,5,2)\), \((1,5,3)\), \((1,6,2)\), \((1,6,4)\), \((1,6,5)\}\),
\(q=7\), \(\,\, \{(0,0,1)\), \((0,1,2)\), \((1,0,0)\), \((1,0,1)\), \((1,0,2)\), \((1,1,4)\), \((1,1,6)\), \((1,2,2)\), \((1,2,6)\), \((1,3,0)\), \((1,3,2)\), \((1,3,4)\), \((1,4,0)\), \((1,4,1)\), \((1,5,6)\), \((1,6,2)\), \((1,6,4)\}\),
\(q=7\), \(\,\, \{(0,0,1)\), \((0,1,0)\), \((0,1,3)\), \((0,1,4)\), \((1,0,0)\), \((1,0,1)\), \((1,0,2)\), \((1,1,0)\), \((1,1,1)\), \((1,1,3)\), \((1,2,0)\), \((1,2,4)\), \((1,2,5)\), \((1,3,0)\), \((1,4,1)\), \((1,4,5)\), \((1,4,6)\), \((1,6,5)\}\),
\(q=7\), \(\,\, \{(0,0,1)\), \((0,1,0)\), \((1,0,0)\), \((1,0,2)\), \((1,0,5)\), \((1,0,6)\), \((1,1,0)\), \((1,1,2)\), \((1,1,6)\), \((1,2,0)\), \((1,2,2)\), \((1,2,6)\), \((1,3,0)\), \((1,3,5)\), \((1,4,5)\), \((1,5,1)\), \((1,5,3)\), \((1,5,4)\}\),
\(q = 9\), \(\,\, \{(0,0,1)\), \((1,0,0)\), \((1,0,\omega )\), \((1,0,\omega ^2)\), \((1,0,2)\), \((1,1,2)\), \((1,1,\omega ^7)\), \((1,\omega ^3,2)\), \((1,\omega ^3,\omega ^6)\), \((1,2,\omega ^3)\), \((1,2,\omega ^5)\), \((1,\omega ^5,0)\), \((1,\omega ^5,1)\), \((1,\omega ^5,2)\),
\((1,\omega ^6,0)\),
\((1,\omega ^6,2)\), \((1,\omega ^6,\omega ^5)\), \((1,\omega ^6,\omega ^7)\}\),
\(q=7\), \(\,\, \{(0,1,0)\), \((0,1,1)\), \((0,1,3)\), \((1,0,3)\), \((1,0,5)\), \((1,0,6)\), \((1,1,1)\), \((1,1,5)\), \((1,3,3)\), \((1,3,6)\), \((1,4,5)\), \((1,4,6)\), \((1,5,0)\), \((1,5,2)\), \((1,5,3)\), \((1,6,1)\), \((1,6,2)\), \((1,6,4)\}\),
\(q=7\), \(\,\, \{(0,0,1)\), \((0,1,1)\), \((0,1,2)\), \((1,0,1)\), \((1,0,2)\), \((1,1,1)\), \((1,1,4)\), \((1,1,6)\), \((1,2,6)\), \((1,3,0)\), \((1,3,2)\), \((1,3,4)\), \((1,4,0)\), \((1,4,1)\), \((1,4,6)\), \((1,5,4)\), \((1,5,6)\), \((1,6,4)\}\),
\(q = 9\), \(\,\, \{(0,0,1)\), \((1,0,0)\), \((1,0,\omega ^2)\), \((1,0,2)\), \((1,1,2)\), \((1,1,\omega ^6)\), \((1,\omega ^3,2)\), \((1,\omega ^3,\omega ^6)\), \((1,\omega ^3,\omega ^7)\), \((1,2,\omega ^3)\), \((1,2,\omega ^5)\), \((1,2,\omega ^6)\), \((1,\omega ^5,0)\), \((1,\omega ^5,1)\),
\((1,\omega ^5,2)\), \((1,\omega ^6,0)\), \((1,\omega ^6,\omega ^5)\), \((1,\omega ^6,\omega ^7)\}\),
\(q=7\), \(\,\, \{(0,0,1)\), \((0,1,0)\), \((0,1,3)\), \((0,1,4)\), \((1,0,0)\), \((1,0,1)\), \((1,0,2)\), \((1,0,5)\), \((1,1,0)\), \((1,1,1)\), \((1,1,3)\), \((1,2,0)\), \((1,2,4)\), \((1,2,5)\), \((1,3,0)\), \((1,4,1)\), \((1,4,5)\), \((1,4,6)\), \((1,6,5)\}\),
\(q = 9\), \(\,\, \{(0,1,1)\), \((1,0,\omega )\), \((1,1,0)\), \((1,1,\omega ^2)\), \((1,1,\omega ^7)\), \((1,\omega ,1)\), \((1,\omega ,\omega ^7)\), \((1,\omega ^2,\omega ^6)\), \((1,\omega ^3,\omega )\), \((1,\omega ^3,\omega ^3)\), \((1,2,\omega ^3)\), \((1,2,\omega ^7)\), \((1,\omega ^5,1)\), \((1,\omega ^6,\omega ^7)\), \((1,\omega ^7,0)\), \((1,\omega ^7,2)\), \((1,\omega ^7,\omega ^5)\), \((1,\omega ^7,\omega ^6)\), \((1,\omega ^7,\omega ^7)\}\),
\(q = 9\), \(\,\, \{(0,1,\omega ^5)\), \((1,0,0)\), \((1,0,\omega )\), \((1,0,\omega ^2)\), \((1,0,2)\), \((1,1,0)\), \((1,1,1)\), \((1,1,2)\), \((1,1,\omega ^5)\), \((1,1,\omega ^7)\), \((1,\omega ,\omega ^7)\), \((1,\omega ^2,\omega ^5)\), \((1,\omega ^5,1)\), \((1,\omega ^5,2)\), \((1,\omega ^6,0)\),
\((1,\omega ^6,2)\), \((1,\omega ^6,\omega ^5)\), \((1,\omega ^6,\omega ^7)\), \((1,\omega ^7,\omega ^3)\}\),
\(q = 9\), \(\,\, \{(0,1,1)\), \((1,0,\omega )\), \((1,1,0)\), \((1,1,\omega ^2)\), \((1,1,\omega ^7)\), \((1,\omega ,1)\), \((1,\omega ,\omega ^7)\), \((1,\omega ^3,\omega )\), \((1,\omega ^3,\omega ^3)\), \((1,\omega ^3,\omega ^6)\), \((1,2,\omega ^3)\), \((1,2,\omega ^6)\), \((1,2,\omega ^7)\), \((1,\omega ^5,1)\), \((1,\omega ^6,\omega ^7)\), \((1,\omega ^7,0)\), \((1,\omega ^7,2)\), \((1,\omega ^7,\omega ^5)\), \((1,\omega ^7,\omega ^7)\}\),
\(q=7\), \(\,\, \{(0,0,1)\), \((0,1,2)\), \((1,0,1)\), \((1,0,4)\), \((1,0,6)\), \((1,1,3)\), \((1,2,0)\), \((1,2,2)\), \((1,2,6)\), \((1,3,0)\), \((1,4,0)\), \((1,4,1)\), \((1,4,2)\), \((1,4,5)\), \((1,5,0)\), \((1,5,2)\), \((1,5,3)\), \((1,6,2)\), \((1,6,4)\), \((1,6,5)\}\),
\(q=13\), \(\,\, \{(0,1,10)\), \((1,0,0)\), \((1,0,3)\), \((1,0,4)\), \((1,1,2)\), \((1,2,8)\), \((1,3,0)\), \((1,4,2)\), \((1,4,7)\), \((1,4,9)\), \((1,5,8)\), \((1,6,0)\), \((1,8,1)\), \((1,9,2)\), \((1,9,5)\), \((1,9,9)\), \((1,9,11)\), \((1,10,1)\), \((1,10,5)\), \((1,10,9)\), \((1,11,1)\}\),
\(q = 9\), \(\,\, \{(0,0,1)\), \((0,1,1)\), \((0,1,\omega )\), \((0,1,\omega ^3)\), \((0,1,2)\), \((0,1,\omega ^6)\), \((1,1,0)\), \((1,1,\omega ^7)\), \((1,\omega ,0)\), \((1,\omega ,1)\), \((1,\omega ^2,0)\), \((1,\omega ^2,\omega )\), \((1,\omega ^3,0)\), \((1,\omega ^3,\omega ^2)\), \((1,2,0)\), \((1,2,\omega ^3)\), \((1,\omega ^5,0)\), \((1,\omega ^5,2)\), \((1,\omega ^6,0)\), \((1,\omega ^6,\omega ^5)\), \((1,\omega ^7,0)\), \((1,\omega ^7,\omega ^6)\}\),
\(q = 9\), \(\,\, \{(0,1,1)\), \((0,1,\omega )\), \((0,1,\omega ^3)\), \((0,1,2)\), \((0,1,\omega ^5)\), \((0,1,\omega ^7)\), \((1,1,\omega ^2)\), \((1,1,\omega ^6)\), \((1,\omega ,\omega ^3)\), \((1,\omega ,\omega ^7)\), \((1,\omega ^2,1)\), \((1,\omega ^2,2)\), \((1,\omega ^3,\omega )\), \((1,\omega ^3,\omega ^5)\),
\((1,2,\omega ^2)\), \((1,2,\omega ^6)\), \((1,\omega ^5,\omega ^3)\), \((1,\omega ^5,\omega ^7)\), \((1,\omega ^6,1)\), \((1,\omega ^6,2)\), \((1,\omega ^7,\omega )\), \((1,\omega ^7,\omega ^5)\}\),
\(q = 9\), \(\,\, \{(0,0,1)\), \((0,1,0)\), \((0,1,1)\), \((0,1,\omega )\), \((0,1,2)\), \((0,1,\omega ^7)\), \((1,1,\omega ^2)\), \((1,1,\omega ^6)\), \((1,\omega ,\omega ^3)\), \((1,\omega ,\omega ^7)\), \((1,\omega ^2,1)\), \((1,\omega ^2,2)\), \((1,\omega ^3,\omega )\), \((1,\omega ^3,\omega ^5)\),
\((1,2,\omega ^2)\), \((1,2,\omega ^6)\), \((1,\omega ^5,\omega ^3)\), \((1,\omega ^5,\omega ^7)\), \((1,\omega ^6,1)\), \((1,\omega ^6,2)\), \((1,\omega ^7,\omega )\), \((1,\omega ^7,\omega ^5)\}\),
\(q = 9\), \(\,\, \{(0,0,1)\), \((0,1,0)\), \((0,1,\omega ^2)\), \((0,1,2)\), \((0,1,\omega ^6)\), \((1,0,0)\), \((1,1,\omega ^3)\), \((1,1,\omega ^7)\), \((1,\omega ,\omega ^2)\), \((1,\omega ,\omega ^6)\), \((1,\omega ^2,\omega )\), \((1,\omega ^2,\omega ^5)\), \((1,\omega ^3,1)\), \((1,\omega ^3,2)\),
\((1,2,\omega ^3)\), \((1,2,\omega ^7)\), \((1,\omega ^5,\omega ^2)\), \((1,\omega ^5,\omega ^6)\), \((1,\omega ^6,\omega )\), \((1,\omega ^6,\omega ^5)\), \((1,\omega ^7,1)\), \((1,\omega ^7,2)\}\),
\(q=11\), \(\,\, \{(0,1,3)\), \((0,1,7)\), \((1,0,4)\), \((1,0,6)\), \((1,1,4)\), \((1,1,6)\), \((1,1,9)\), \((1,1,10)\), \((1,2,4)\), \((1,2,6)\), \((1,2,8)\), \((1,3,4)\), \((1,3,10)\), \((1,4,0)\), \((1,4,4)\), \((1,4,10)\), \((1,5,6)\), \((1,6,2)\), \((1,6,3)\), \((1,6,5)\), \((1,7,10)\), \((1,8,4)\), \((1,9,0)\), \((1,9,1)\), \((1,10,7)\}\),
\(q=13\), \(\,\, \{(0,1,9)\), \((1,0,4)\), \((1,1,10)\), \((1,1,11)\), \((1,1,12)\), \((1,2,7)\), \((1,2,8)\), \((1,2,11)\), \((1,3,7)\), \((1,4,7)\), \((1,5,5)\), \((1,6,0)\), \((1,6,6)\), \((1,6,8)\), \((1,7,6)\), \((1,7,8)\), \((1,7,10)\), \((1,8,5)\), \((1,9,4)\), \((1,9,9)\), \((1,9,12)\), \((1,10,5)\), \((1,10,11)\), \((1,10,12)\), \((1,11,0)\), \((1,11,3)\), \((1,11,10)\), \((1,12,0)\), \((1,12,3)\), \((1,12,6)\}\),
\(q=13\), \(\,\, \{(0,1,7)\), \((1,0,4)\), \((1,0,6)\), \((1,0,9)\), \((1,1,2)\), \((1,2,5)\), \((1,2,8)\), \((1,3,8)\), \((1,4,3)\), \((1,4,7)\), \((1,4,10)\), \((1,5,3)\), \((1,6,0)\), \((1,6,5)\), \((1,7,7)\), \((1,7,10)\), \((1,7,12)\), \((1,8,12)\), \((1,9,0)\), \((1,9,4)\), \((1,9,6)\), \((1,9,9)\), \((1,9,11)\), \((1,9,12)\), \((1,10,0)\), \((1,10,5)\), \((1,10,10)\), \((1,11,1)\), \((1,12,6)\), \((1,12,8)\), \((1,12,9)\}\),
\(q=29\), \(\,\, \{(0,1,6)\), \((0,1,23)\), \((1,0,0)\), \((1,0,1)\), \((1,0,2)\), \((1,1,0)\), \((1,1,1)\), \((1,1,22)\), \((1,2,0)\), \((1,2,23)\), \((1,2,26)\), \((1,3,22)\), \((1,3,23)\), \((1,3,26)\), \((1,4,15)\), \((1,5,6)\), \((1,5,7)\), \((1,5,21)\), \((1,6,21)\), \((1,7,12)\), \((1,7,13)\), \((1,7,19)\), \((1,8,5)\), \((1,10,9)\), \((1,10,12)\), \((1,10,15)\), \((1,11,0)\), \((1,11,2)\), \((1,11,12)\), \((1,12,1)\), \((1,13,2)\),
\((1,13,4)\), \((1,13,10)\), \((1,13,27)\), \((1,15,15)\), \((1,16,8)\), \((1,17,23)\), \((1,19,28)\), \((1,20,16)\), \((1,22,11)\), \((1,25,20)\), \((1,25,22)\), \((1,25,24)\), \((1,26,3)\), \((1,26,21)\}\).
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Cuntz, M., Geis, D. Combinatorial simpliciality of arrangements of hyperplanes. Beitr Algebra Geom 56, 439–458 (2015). https://doi.org/10.1007/s13366-014-0190-x
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DOI: https://doi.org/10.1007/s13366-014-0190-x