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Polytopes of high rank for the symmetric groups

par Fernandes, Maria Elisa;Leemans, Dimitri
Référence Advances in mathematics, 228, 6, page (3207-3222)
Publication Publié, 2011

Article révisé par les pairs

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Résumé :

In the Atlas of abstract regular polytopes for small almost simple groups by Leemans and Vauthier, the polytopes whose automorphism group is a symmetric group Sn of degree 5≤n≤9 are available. Two observations arise when we look at the results: (1) for n≥5, the (n-1)-simplex is, up to isomorphism, the unique regular (n-1)-polytope having Sn as automorphism group and, (2) for n≥7, there exists, up to isomorphism and duality, a unique regular (n-2)-polytope whose automorphism group is Sn. We prove that (1) is true for n≠4 and (2) is true for n≥7. Finally, we also prove that Sn acts regularly on at least one abstract polytope of rank r for every 3≤r≤n-1. © 2011 Elsevier Inc.