The number of string C-groups of high rank (Q6608014)

A pair \(\Gamma := (G,S)\) is a \emph{string C-group of rank \(r\)} if the following conditions hold:\N\begin{itemize}\N\item[1.] \(G\) is a group, \(S:=\{\rho_0,\dots, \rho_{r-1}\}\) is an ordered set of involutions such that \(G=\langle S\rangle\).\N\item[2.] For all \(i,j\in\{0,\dots, r-1\}\), \(|i-j|>1\) implies that \((\rho_i\rho_j)^2=1\).\N\item[3.] For all \(J, K \subseteq\{0,\dots, r-1\}\), \(\langle \rho_j \mid j\in J\rangle \cap \langle \rho_k \mid k\in K\rangle = \langle \rho_j \mid j \in J\cap K\rangle\).\N\end{itemize}\N\NOne of the things that makes string C-groups interesting is that they are in one-to-one correspondence with abstract regular polytopes. This paper concerns string C-groups for which \(G\) is a subgroup of \(S_n\), a finite symmetric group.\N\NIt is known that if \(G\) is transitive and \(\Gamma\) has rank \(r\geq (n+3)/2\), then \(G\) is necessarily the symmetric group \(S_n\). It has been conjectured that if \(n\) is large enough, then, up to isomorphism and duality, the number of string C-groups of rank \(r\) with \(G=S_n\) and \(r\geq (n+3)/2\), is the same as the number of string C-groups of rank \(r+1\) with \(G=S_{n+1}\).\N\NThe authors prove this conjecture. More precisely, they prove, that for each fixed integer \(\kappa\geq 1\), there exists an integer \(c_\kappa\) such that, for all \(n\geq 2\kappa+3\), the number of string C-groups of rank \(n-\kappa\) with \(G=S_n\) is, up to isomorphism and duality, equal to \(c_\kappa\).\N\NA consequence of this result is the complete classification of all string C-groups of \(S_n\) with rank \(n-\kappa\) for \(\kappa\in\{1,\dots, 7\}\) and \(n\geq 2\kappa+3\).