Remark. Distinguished generators and the intersection property [frap-000I]

✍️sourceAGENTDRAFTED

Choose a base flag \(\Phi \) in a regular abstract polytope \(P\) of rank \(n\). By flag freeness, there is a unique automorphism \(\rho _i\) taking \(\Phi \) to its rank-\(i\) neighbor, for \(0\leq i<n\). The diamond condition and uniqueness give \(\rho _i^2=1\). For \(|i-j|>1\), the two rank changes are independent; \(\rho _i\rho _j\) and \(\rho _j\rho _i\) agree on \(\Phi \), hence are equal. Connectivity shows that the \(\rho _i\) reach every flag, so they generate \(\Gamma (P)\). These are the distinguished generators of [schulte2017regular, Section 5, pp. 13--14].

For \(I\subseteq \{0,\ldots ,n-1\}\), put \(G_I=\langle \rho _i:i\in I\rangle \). They also satisfy

This intersection property is part of the string C-group description [schulte2017regular, Section 5, equation (18)]. Involution and nonadjacent commutation relations alone do not reconstruct a polytope; the group-theoretic characterization includes this additional condition [schulte2017regular, Section 5, p. 14].