Arrangements with symmetries
Final Report Abstract
This project was about arrangements of hyperplanes and had two major goals: to understand the structure of the special simplicial arrangements of hyperplanes, including classifications, and to apply the results to construct a counterexample to the longstanding conjecture of Terao. Combining group theory, computational enumerations, and traditional methods of geometry we achieved significant results for all objectives: Concerning the class of simplicial arrangements, we gave a complete classification of supersolvable simplicial arrangements, we obtained partial results on the more general (infinite) Tits arrangements, and found a computer free proof for the finiteness of the number of crystallographic arrangements in each rank. Combining the properties of simpliciality and freeness, David Geis worked on both goals simultaneously in his dissertation. His most important theorems yield, under some mild assumptions, bounds for the number of arrangements of lines for which the roots of the characteristic polynomial are only real numbers. As a corollary he obtains a new proof of the Dirac Motzkin Conjecture for this special class. Regarding Terao’s Conjecture, we proved that all ideal arrangements in Weyl arrangements are inductively free and introduced a new class of free arrangements based on the multiple addition theorem. We did not succeed in finding a counterexample. However, the new methods implemented for this search produced for example unknown (n4) configurations of lines. A very successful workshop on “Hyperplane Arrangements and Reflection Groups” which took place in Hannover in September 2019 was funded by this project.
Publications
- Tits arrangements on cubic curves (2017)
Michael Cuntz, David Geis
- (22_4) and (26_4) configurations of lines. Ars Math. Contemp. 14, no. 1 (2018), 157–163
Michael Cuntz
(See online at https://doi.org/10.26493/1855-3974.1402.733) - Combinatorics of free and simplicial line arrangements. (2018)
David Geis
- On the combinatorics of Tits arrangements. Dissertation (2018)
David Geis
(See online at https://dx.doi.org/10.15488/3483) - A bound for crystallographic arrangements. (2019)
Michael Cuntz
- Arrangements of ideal type are inductively free. Internat. J. Algebra Comput. 29 (2019), no. 5, 761–773
Michael Cuntz, Gerhard Roehrle, Anne Schauenburg
(See online at https://doi.org/10.1142/S0218196719500267) - On simplicial arrangements in P3 (R) with splitting polynomial. (2019)
David Geis
- On the Tits cone of a Weyl groupoid. Comm. Algebra. 47, no. 12 (2019), 5261–5285
Michael Cuntz, Bernhard Mühlherr, Christian J. Weigel
(See online at https://doi.org/10.1080/00927872.2019.1617873) - Supersolvable simplicial arrangements. Adv. in Appl. Math. 107 (2019), 32–73
Michael Cuntz, Paul Mücksch
(See online at https://doi.org/10.1016/j.aam.2019.02.008) - MAT-free reflection arrangements. Electron. J. Combin. 27, no. 1 (2020), P1.28
Michael Cuntz, Paul Mücksch
(See online at https://doi.org/10.37236/8820)
