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

TL;DR

The paper classifies transitive proper subgroups $G$ of $S_n$ with C-rank at least $n/2$ for $n \ge 14$, focusing on the case where the rank is exactly $n/2$. It develops the framework of independent generating sets, SgGi permutation representation graphs, and string C-groups to study imprimitive actions, proving that, up to duality, only certain wreath-product and block-structure groups can occur: $C_2\times S_{n/2}$, $(C_2)^{n/2-1}:S_{n/2}$, $(C_2)^{n/2}:S_{n/2}$, or $(S_{n/2})^2:C_2$. The analysis splits into two imprimitive cases—blocks of size $2$ and two blocks of size $n/2$—and yields a comprehensive list of possible permutation representation graphs (supported by computational checks) that realize these string C-group structures, thereby identifying the corresponding regular polytopes of rank $r=n/2$. The work extends prior results on maximal ranks (noting that reducing rank to $n/2$ substantially increases the number of polytopes) and provides a foundation for further verification of the intersection property and a complete catalog of the resulting polytopes. The findings have implications for understanding the symmetry and combinatorial structure of abstract polytopes associated with transitive subgroups of $S_n$.

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.

Theorems & Definitions (86)

• Theorem 1.1
• Conjecture 1.2
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.7
• proof
• Lemma 2.8