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A Property of Isometric Mappings Between Dual Polar Spaces of Type DQ(2n,K)
Bart De Bruyn
Department of Mathematics, Ghent University, Krijgslaan 281 (S22), B-9000 Gent, Belgium
Annals of Combinatorics 14 (3) pp.307-318 September, 2010
AMS Subject Classification: 51A45, 51A50
Let f be an isometric embedding of the dual polar space Δ= DQ(2n,K) into Δ' = DQ(2n,K'). Let P denote the point-set of Δ and let e': Δ' → ∑' PG(2n-1,K') denote the spin-embedding of Δ. We show that for every locally
singular hyperplane H of Δ, there exists a unique locally singular hyperplane H' of Δ' such that f(H)=f(P) ∩ H'. We use this to show that there exists a subgeometry ∑ PG(2n-1,K) of ∑' such that: (i) e' ° f(x) ∈ ∑ for every point x of Δ; (ii) e := e' ° f defines a full embedding of Δ into ∑, which is isomorphic to the spin-embedding of ∑.
Keywords: composition, probability, lattice paths, fence poset, asymptotics


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