Characterization of isometric embeddings of Grassmann graphs

Abstract

Abstract Let V be an n-dimensional left vector space over a division ring R. We write Gk(V ) for the Grassmannian formed by k-dimensional subspaces of V and denote by Γk(V ) the associated Grassmann graph. Let also V′ be an n0-dimensional left vector space over a division ring R0. Isometric embeddings of Γk(V ) in Γk′(V′) are classified in [13]. A classification of J(n; k)-subsets in Gk′(V′), i.e. the images of isometric embeddings of the Johnson graph J(n; k) in Γk′(V′), is presented in [12]. We characterize isometric embeddings of Γk(V ) in Γk′(V′) as mappings which transfer apartments of Gk(V) to J(n; k)-subsets of Gk′(V′). This is a generalization of the earlier result concerning apartment preserving mappings [11, Theorem 3.10].