Isometric embeddings of half-cube graphs in half-spin Grassmannians (1106.5435v2)

Abstract: Let $\Pi$ be a polar space of type $\textsf{D}{n}$. 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 $\textsf{D}_{n'}$ under the assumption that $n\ge 6$ is even.