Metric characterization of apartments in dual polar spaces

摘要

Let Π be a polar space of rank n and let G_k(Π), kε{0,...,n-1} be the polar Grassmannian formed by k-dimensional singular subspaces of Π. The corresponding Grassmann graph will be denoted by Γ_k(Π). We consider the polar Grassmannian G_(n-1)(Π) formed by maximal singular subspaces of π and show that the image of every isometric embedding of the n-dimensional hypercube graph Hn in Γ_(n-1)(Π) is an apartment of G_(n-1)(Π). This follows from a more general result concerning isometric embeddings of H_m, m≤n in Γ_(n-1)(Π). As an application, we classify all isometric embeddings of Γ_(n-1)(Π) in Γ_(n′-1)(Π′), where Π′ is a polar space of rank n′≥n.