Rank reduction of string C-group representations
HTML articles powered by AMS MathViewer
- by Peter A. Brooksbank and Dimitri Leemans PDF
- Proc. Amer. Math. Soc. 147 (2019), 5421-5426 Request permission
Abstract:
We show that a rank reduction technique for string C-group representations first used in [Adv. Math. 228 (2018), pp. 3207–3222] for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on $d$-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks $3\leqslant n\leqslant d$. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group $\operatorname {Alt}(11)$—the only known group having “rank gaps”—is perhaps more unusual than previously thought.References
- P. A. Brooksbank, J. T. Ferrara, and D. Leemans, Orthogonal groups in characteristic 2 acting on polytopes of high rank, Discrete Comput. Geom. (2019), https://doi.org/10.1007/s00454-019-00083-0.
- Thomas Connor and Dimitri Leemans, C-groups of Suzuki type, J. Algebraic Combin. 42 (2015), no. 3, 849–860. MR 3403184, DOI 10.1007/s10801-015-0605-2
- Maria Elisa Fernandes and Dimitri Leemans, Polytopes of high rank for the symmetric groups, Adv. Math. 228 (2011), no. 6, 3207–3222. MR 2844941, DOI 10.1016/j.aim.2011.08.006
- M. E. Fernandes and D. Leemans, String C-group representations of alternating groups, preprint, arXiv 1810.12450, 2018.
- Maria Elisa Fernandes, Dimitri Leemans, and Mark Mixer, Polytopes of high rank for the alternating groups, J. Combin. Theory Ser. A 119 (2012), no. 1, 42–56. MR 2844081, DOI 10.1016/j.jcta.2011.07.006
- Peter McMullen and Egon Schulte, Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge University Press, Cambridge, 2002. MR 1965665, DOI 10.1017/CBO9780511546686
- B. Monson and Egon Schulte, Reflection groups and polytopes over finite fields. I, Adv. in Appl. Math. 33 (2004), no. 2, 290–317. MR 2074400, DOI 10.1016/j.aam.2003.11.002
- B. Monson and Egon Schulte, Reflection groups and polytopes over finite fields. II, Adv. in Appl. Math. 38 (2007), no. 3, 327–356. MR 2301701, DOI 10.1016/j.aam.2005.12.001
- B. Monson and Egon Schulte, Reflection groups and polytopes over finite fields. III, Adv. in Appl. Math. 41 (2008), no. 1, 76–94. MR 2419764, DOI 10.1016/j.aam.2007.07.001
Additional Information
- Peter A. Brooksbank
- Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
- MR Author ID: 321878
- Email: pbrooksb@bucknell.edu
- Dimitri Leemans
- Affiliation: Département de Mathématique, C.P. 216 Algèbre et Combinatoire, Université Libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles, Belgium
- MR Author ID: 613090
- ORCID: 0000-0002-4439-502X
- Email: dleemans@ulb.ac.be
- Received by editor(s): December 3, 2018
- Received by editor(s) in revised form: March 28, 2019
- Published electronically: July 1, 2019
- Additional Notes: This work was partially supported by a grant from the Simons Foundation (#281435 to the first author) and by the Hausdorff Research Institute for Mathematics
- Communicated by: Pham Huu Tiep
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 5421-5426
- MSC (2010): Primary 52B11, 20D06
- DOI: https://doi.org/10.1090/proc/14666
- MathSciNet review: 4021100