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

Let $Π$ be a polar space of type $\textsf{D}_{n}$. Denote by ${\mathcal G}_δ(Π)$, $δ\in \{+,-\}$ the associated half-spin Grassmannians and write $Γ_δ(Π)$ for the corresponding half-spin Grassmann graphs. In the case when $n\ge 4$ is even, the apartments of ${\mathcal G}_δ(Π)$ will be characterized as the images of isometric embeddings of the half-cube graph $\frac{1}{2}H_n$ in $Γ_δ(Π)$. As an application, we describe all isometric embeddings of $Γ_δ(Π)$ 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.

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