Metric characterization of apartments in dual polar spaces |
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Authors: | Mark Pankov |
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Institution: | Department of Mathematics and Informatics, University of Warmia and Mazury, ?olnierska 14A, 10-561 Olsztyn, Poland |
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Abstract: | Let Π be a polar space of rank n and let Gk(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 Γk(Π). We consider the polar Grassmannian Gn−1(Π) formed by maximal singular subspaces of Π and show that the image of every isometric embedding of the n-dimensional hypercube graph Hn in Γn−1(Π) is an apartment of Gn−1(Π). This follows from a more general result concerning isometric embeddings of Hm, m?n in Γn−1(Π). As an application, we classify all isometric embeddings of Γn−1(Π) in Γn′−1(Π′), where Π′ is a polar space of rank n′?n. |
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Keywords: | Apartment Dual polar space Hypercube graph Isometric embedding |
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