Characterization of isometric embeddings of Grassmann graphs

2012/03/01 by Pankov, Mark
#15A04 #51M35 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1203.0105

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 Γ_k(V) the associated Grassmann graph. Let also V' be an n'-dimensional left vector space over a division ring R'. Isometric embeddings of Γ_k(V) in Γ_k'(V') are classified in \citePankov2. 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 Γ_k'(V'), is presented in \citePankov1. We characterize isometric embeddings of Γ_k(V) in Γ_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.

Related

• Embeddings of Grassmann graphs
• Isometric embeddings of Johnson graphs in Grassmann graphs
• Isometric embeddings of half-cube graphs in half-spin Grassmannians
• Metric characterization of apartments in dual polar spaces
• Isometric embeddings of polar Grassmannians and metric characterizations of their apartments