A Property of Isometric Mappings Between Dual Polar Spaces of Type DQ(2n, K)

作者：De Bruyn Bart^*

来源：Annals of Combinatorics, 2010, 14(3): 307-318.

DOI：10.1007/s00026-010-0061-6

• isometric embedding
• dual polar space
• hyperplane
• spin-embedding

Let f be an isometric embedding of the dual polar space Delta = DQ(2n, K) into Delta' = DQ(2n, K'). Let P denote the point-set of Delta and let e' : Delta' -> Sigma' congruent to PG(2(n) - 1, K') denote the spin-embedding of Delta'. We show that for every locally singular hyperplane H of Delta, there exists a unique locally singular hyperplane H' of Delta' such that f(H) = f(P) boolean AND H'. We use this to show that there exists a subgeometry Sigma congruent to PG(2(n) - 1, K) of Sigma' such that: (i) e' circle f (x) is an element of Sigma for every point x of Delta; (ii) e := e' circle f defines a full embedding of Delta into Sigma, which is isomorphic to the spin-embedding of Delta.

• 出版日期2010-9