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

TL;DR

This paper characterizes apartments in half-spin Grassmannians of even-dimensional polar spaces as isometric embeddings of half-cube graphs and describes all such embeddings between related Grassmann graphs.

Contribution

It provides a new characterization of apartments via isometric embeddings and classifies all such embeddings between half-spin Grassmann graphs.

Findings

Apartments are characterized as images of half-cube graph embeddings.

All isometric embeddings between certain Grassmann graphs are described.

Results apply to polar spaces of type D with even dimensions.

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.