Characterization of isometric embeddings of Grassmann graphs

TL;DR

This paper characterizes isometric embeddings of Grassmann graphs by showing they map apartments to Johnson graph subsets, extending previous results on apartment-preserving mappings.

Contribution

It generalizes earlier classifications by characterizing isometric embeddings as those that map apartments to Johnson graph subsets.

Findings

Classified isometric embeddings of Grassmann graphs.

Connected embeddings with Johnson graph subsets.

Extended previous apartment-preserving mapping results.

Abstract

Let $V$ be an $n$-dimensional left vector space over a division ring $R$. We write ${\mathcal G}_{k}(V)$ for the Grassmannian formed by $k$-dimensional subspaces of $V$ and denote by $\Gamma_{k}(V)$ the associated Grassmann graph. Let also $V'$ be an $n'$-dimensional left vector space over a division ring $R'$. Isometric embeddings of $\Gamma_{k}(V)$ in $\Gamma_{k'}(V')$ are classified in \cite{Pankov2}. A classification of $J(n,k)$-subsets in ${\mathcal G}_{k'}(V')$, i.e. the images of isometric embeddings of the Johnson graph $J(n,k)$ in $\Gamma_{k'}(V')$, is presented in \cite{Pankov1}. We characterize isometric embeddings of $\Gamma_{k}(V)$ in $\Gamma_{k'}(V')$ as mappings which transfer apartments of ${\mathcal G}_{k}(V)$ to $J(n,k)$-subsets of ${\mathcal G}_{k'}(V')$. This is a generalization of the earlier result concerning apartments preserving mappings \cite[Theorem 3.10]{Pankov-book}.