<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 13 Issue 3" %>
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
bdb@cage.ugent.be
Annals of Combinatorics 14 (3) pp.307-318 September, 2010
AMS Subject Classification: 51A45, 51A50
Abstract:
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

References:

1. Buekenhout, F., Cameron, P.J.: Projective and affine geometry over division rings. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry, pp. 27--62. North-Holland, Amsterdam (1995)

2. Cameron, P.J.: Dual polar spaces. Geom. Dedicata 12(1), 75--85 (1982)

3. Cardinali, I., De Bruyn, B., Pasini, A.: Minimal full polarized embeddings of dual polar spaces. J. Algebraic Combin. 25(1), 7--23 (2007)

4. Chevalley, C.C.: The Algebraic Theory of Spinors. Columbia University Press, New York (1954)

5. De Bruyn, B.: The hyperplanes of DQ(2n, K) and DQ-(2n+1,q) which arise from their spin-embeddings. J. Combin. Theory Ser. A 114(4), 681--691 (2007)

6. De Bruyn, B.: The structure of the spin-embeddings of dual polar spaces and related geometries. European J. Combin. 29(5), 1242--1256 (2008)

7. De Bruyn, B.: Two classes of hyperplanes of dual polar spaces without subquadrangular quads. J. Combin. Theory Ser. A 115(5), 893--902 (2008)

8. De Bruyn, B., Pasini, A.: Minimal scattered sets and polarized embeddings of dual polar spaces. European J. Combin. 28(7), 1890--1909 (2007)

9. Shult, E.E.: On Veldkamp lines. Bull. Belg. Math. Soc. Simon Stevin 4(2), 299--316 (1997)

10. Shult, E.E., Thas, J.A.: Hyperplanes of dual polar spaces and the spin module. Arch. Math. (Basel) 59(6), 610--623 (1992)