Isometric Embeddings of Half-Cube Graphs in Half-Spin Grassmannians

Let Π be a polar space of type Dn. Denote by Gδ(Π), δ ∈ {+,−} the associated half-spin Grassmannians and write Γδ(Π) for the corresponding half-spin Grassmann graphs. In the case when n ≥ 4 is even, the apartments of Gδ(Π) will be characterized as the images of isometric embeddings of the half-cube graph 1 2 Hn in Γδ(Π). As an application, we describe all isometric embeddings of Γδ(Π) in the half-spin Grassmann graphs associated to a polar space of type Dn′ under the assumption that n ≥ 6 is even.