Accepted to Advances in Geometry CHARACTERIZATION OF ISOMETRIC EMBEDDINGS OF GRASSMANN GRAPHS
نویسنده
چکیده
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 n′-dimensional left vector space over a division ring R′. 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].
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