Two gluing methods for string C-group representations of the symmetric groups

TL;DR

The paper investigates high-rank string C-group representations of $S_n$ by examining permutation representation graphs and testing Cameron, Fernandes and Leemans's conjecture that certain graphs are CPR graphs. It introduces two gluing techniques for CPR graphs, proving a successful gluing theorem (Theorem \text{'taping'}) that often yields new CPR graphs and related symmetric-group structures, while showing that a second, seemingly natural gluing approach fails in general. A key outcome is the construction of a counterexample (graph (X)) demonstrating the conjecture's failure, though the methods succeed in special cases such as simplex-based graphs. The work provides constructive tools for building and ruling out high-rank string C-group representations of $S_n$, with potential implications for enumerating such representations and relating to known integer sequences.

Abstract

The study of string C-group representations of rank at least $n/2$ for the symmetric group $S_n$ has gained a lot of attention in the last fifteen years. In a recent paper, Cameron et al. gave a list of permutation representation graphs of rank $r\geq n/2$ for $S_n$, having a fracture graph and a non-perfect split. They conjecture that these graphs are permutation representation graphs of string C-groups. In trying to prove this conjecture, we discovered two new techniques to glue two CPR graphs for symmetric groups together. We discuss the cases in which they yield new CPR graphs. By doing so, we invalidate the conjecture of Cameron et al. We believe our gluing techniques will be useful in the study of string C-group representations of high ranks for the symmetric groups.

Theorems & Definitions (31)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 2.4
• Proposition 2.5
• Lemma 2.6
• proof
• Theorem 2.7
• Lemma 3.1