Metric characterization of apartments in dual polar spaces

2010/09/10 by Mark Pankov, Pankov, Mark
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1009.1997

arxiv created 2010/09/14 · arxiv updated 2010/09/15

Abstract

Let Π be a polar space of rank n and let \mathcal G_k(Π), k∈ \0,…,n-1\ be the polar Grassmannian formed by k-dimensional singular subspaces of Π. The corresponding Grassmann graph will be denoted by Γ_k(Π). We consider the polar Grassmannian \mathcal G_n-1(Π) formed by maximal singular subspaces of Π and show that the image of every isometric embedding of the n-dimensional hypercube graph H_n in Γ_n-1(Π) is an apartment of \mathcal G_n-1(Π). This follows from a more general result (Theorem 2) concerning isometric embeddings of H_m, m≤ n in Γ_n-1(Π). As an application, we classify all isometric embeddings of Γ_n-1(Π) in Γ_n'-1(Π'), where Π' is a polar space of rank n'≥ n (Theorem 3).

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