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

TL;DR

This paper investigates regular polytopes of rank exactly half the degree for transitive groups, revealing that lowering the rank from the maximum significantly increases the number of such polytopes.

Contribution

It extends previous work by analyzing the case where the rank equals half the degree, showing a substantial increase in regular polytopes when the rank is reduced by one.

Findings

Only two polytopes attain the maximal rank when $n/2$ is odd for $n extgreater=12$.

Reducing the rank from the maximum to $n/2$ increases the number of regular polytopes.

The study provides new insights into the structure of regular polytopes with specific automorphism groups.

Abstract

Previous research established that the maximal rank of the abstract regular polytopes whose automorphism group is a transitive proper subgroup of $\mbox{S}_n$ is $n/2 + 1$. Up to isomorphism and duality, when $n\geq 12$, 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 $n/2$ ($n\geq 14$). Our analysis suggests that reducing the rank by one results in a substantial increase in the number of regular polytopes.