Metric characterization of apartments in dual polar spaces

Abstract

Let $\Pi$ be a polar space of rank $n$ and let ${\mathcal{G}}_k(\Pi )$, $k\in \{0,\dots ,n-1\}$ be the polar Grassmannian formed by $k$-dimensional singular subspaces of $\Pi$. The corresponding Grassmann graph will be denoted by ${\Gamma }_k(\Pi )$. We consider the polar Grassmannian ${\mathcal{G}}_{n-1}(\Pi )$ formed by maximal singular subspaces of $\Pi$ and show that the image of every isometric embedding of the $n$-dimensional hypercube graph $H_n$ in ${\Gamma }_{n-1}(\Pi )$ is an apartment of ${\mathcal{G}}_{n-1}(\Pi )$. This follows from a more general result (Theorem 2) concerning isometric embeddings of $H_m$, $m\le n$ in ${\Gamma }_{n-1}(\Pi )$. As an application, we classify all isometric embeddings of ${\Gamma }_{n-1}(\Pi )$ in ${\Gamma }_{n^{\prime }-1}({\Pi }^{\prime })$, where ${\Pi }^{\prime }$ is a polar space of rank $n^{\prime }\ge n$ (Theorem 3).