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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An extension of the classification of high rank regular polytopes


Authors: Maria Elisa Fernandes, Dimitri Leemans and Mark Mixer
Journal: Trans. Amer. Math. Soc. 370 (2018), 8833-8857
MSC (2010): Primary 52B11, 20D06
DOI: https://doi.org/10.1090/tran/7425
Published electronically: September 13, 2018
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Abstract: Up to isomorphism and duality, there are exactly two nondegenerate abstract regular polytopes of rank greater than $ n-3$ (one of rank $ n-1$ and one of rank $ n-2$) with automorphism groups that are transitive permutation groups of degree $ n\geq 7$. In this paper we extend this classification of high rank regular polytopes to include the ranks $ n-3$ and $ n-4$. The result is, up to isomorphism and duality, there are exactly seven abstract regular polytopes of rank $ n-3$ for each $ n\geq 9$, and there are nine abstract regular polytopes of rank $ n-4$ for each $ n \geq 11$. Moreover, we show that if a transitive permutation group $ \Gamma $ of degree $ n \geq 11$ is the automorphism group of an abstract regular polytope of rank at least $ n-4$, then $ \Gamma \cong S_n$.


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Additional Information

Maria Elisa Fernandes
Affiliation: Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Email: maria.elisa@ua.pt

Dimitri Leemans
Affiliation: Université Libre de Bruxelles, Département de Mathématique, C.P.216 - Algèbre et Combinatoire, Boulevard du Triomphe, 1050 Brussels, Belgium
Email: dleemans@ulb.ac.be

Mark Mixer
Affiliation: Department of Applied Mathematics, Wentworth Institute of Technology, Boston, Massachusetts 02115
Email: mixerm@wit.edu

DOI: https://doi.org/10.1090/tran/7425
Keywords: Abstract regular polytopes, string C-groups, permutation groups
Received by editor(s): September 5, 2017
Received by editor(s) in revised form: October 4, 2017
Published electronically: September 13, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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