Metric characterization of apartments in dual polar spaces

TL;DR

This paper characterizes apartments in dual polar spaces through metric properties, showing that isometric embeddings of hypercube graphs correspond to apartments, and classifies embeddings between different polar Grassmannians.

Contribution

It establishes a metric characterization of apartments in dual polar spaces and classifies isometric embeddings between their Grassmann graphs.

Findings

Isometric embeddings of hypercube graphs map to apartments.

Every isometric embedding of H_n in the Grassmann graph corresponds to an apartment.

Classifies all isometric embeddings between Grassmann graphs of different 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 $\Gamma_{n'-1}(\Pi')$, where $\Pi'$ is a polar space of rank $n'\ge n$ (Theorem 3).