Characterization of apartments in polar Grassmannians (Q431543)

Let \(\Pi\) be a polar space of rank \(n\geq\,3\) and let \(\Gamma_k(\Pi)\) be the Grassmann graph formed by the \(k\)-dimensional subspaces of \(\Pi\). The author investigates the embeddings of the polar Johnson graph \(PJ(l,m)\) in \(\Gamma_k(\Pi)\) such that \(m\leq\,k\leq\,n-2\) and \(l-m\leq\,n-k\). In the case with \(l-m=n-k\) the author gets a characterization of apartments in parabolic subspaces of the associated Grassmann space. As application the author shows that parabolic subspaces can be characterized as convex subspaces isomorphic to polar Grassmann spaces.