Regular polytopes of rank n/2 for transitive groups of degree n

2024/07/22 by Fernandes, Maria Elisa, Piedade, Claudio Alexandre
#05C25 #20B30 #20B35 #52B11 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2407.16003

Abstract

Previous research established that the maximal rank of the abstract regular polytopes whose automorphism group is a transitive proper subgroup of S_n is n/2 + 1. Up to isomorphism and duality, when n≥ 12, there are only two polytopes attaining this rank and they occur when n/2 is odd, and hence have even rank. In this paper, we investigate the case where the rank is equal to n/2 (n≥ 14). Our analysis suggests that reducing the rank by one results in a substantial increase in the number of regular polytopes.

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