Metric characterization of apartments in dual polar spaces

Details Metric characterization of apartments in dual polar spaces Metric characterization of apartments in dual polar spaces

2010-09-10

arxiv.org/abs/1009.1997v2

Mathematics

Combinatorics

Scientific paper

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).

Affiliated with

Also associated with

No associations

Rating

Metric characterization of apartments in dual polar spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Metric characterization of apartments in dual polar spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Metric characterization of apartments in dual polar spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-672492