Isometric embeddings of half-cube graphs in half-spin Grassmannians

Abstract

Let $\Pi$ be a polar space of type ${\text{D}}_n$. Denote by ${\mathcal{G}}_{\delta }(\Pi )$, $\delta \in \{+,-\}$ the associated half-spin Grassmannians and write ${\Gamma }_{\delta }(\Pi )$ for the corresponding half-spin Grassmann graphs. In the case when $n\ge 4$ is even, the apartments of ${\mathcal{G}}_{\delta }(\Pi )$ will be characterized as the images of isometric embeddings of the half-cube graph $\frac{1}{2}{H}_n$ in ${\Gamma }_{\delta }(\Pi )$. As an application, we describe all isometric embeddings of ${\Gamma }_{\delta }(\Pi )$ in the half-spin Grassmann graphs associated to a polar space of type ${\text{D}}_{n^{\prime }}$ under the assumption that $n\ge 6$ is even.