Polytopes of high rank for the symmetric groups

By Maria Elisa Fernandes, Dimitri Leemans. 2011

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Abstract 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 S n 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 S n 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 S n . We prove that (1) is true for n ≠ 4 and (2) is true for n ⩾ 7 . Finally, we also prove that S n acts regularly on at least one abstract polytope of rank r for every 3 ⩽ r ⩽ n − 1 .

Subfields: Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/67be92eca2e92e207b9d04dbcd6a9fbba3b87b4c