Combinatorial Theory

Let \(\Omega_i\) and \(\Omega_j\) be the sets of elements of respective types \(i\) and \(j\) of a polar space \(\Delta\) of rank at least \(3\). We show that a permutation \(\rho\) of \(\Omega_i \cup \Omega_j\) with the property that, for each \(I \in \Omega_i \) and \(J\in\Omega_j\), \(I\) and \(J\) generate a maximal singular subspace in \(\Delta\) if and only if \(\rho(I)\) and \(\rho(J)\) generate a maximal singular subspace in \(\Delta\), is induced by an automorphism of \(\Delta\). Building-theoretically, this means that if \(\rho\) preserves a certain Weyl distance in the Tits-building corresponding to \(\Delta\), then it preserves all Weyl-distances.

Mathematics Subject Classifications: 51E24, 51A50

Keywords: Polar spaces, Weyl distance