A decomposition of the natural embedding spaces for the symplectic dual polar spaces

Abstract

Let e be the Grassmann-embedding of the symplectic dual polar space DW ( 2 n - 1 , K ) into PG( W), where W is a 2 n n - 2 n n - 2 -dimensional vector space over K . For every point z of DW ( 2 n - 1 , K ) and every i ∈ N , Δ i ( z ) denotes the set of points at distance i from z. We show that for every pair { x , y } of mutually opposite points of DW ( 2 n - 1 , K ) , W can be written as a direct sum W 0 ⊕ W 1 ⊕ ⋯ ⊕ W n such that the following four properties hold for every i ∈ { 0 , … , n } : (1) 〈 e ( Δ i ( x ) ∩ Δ n - i ( y ) ) 〉 = PG ( W i ) ; (2) 〈 e ( ⋃ j ⩽ i Δ j ( x ) ) 〉 = PG ( W 0 ⊕ W 1 ⊕ ⋯ ⊕ W i ) ; (3) 〈 e ( ⋃ j ⩽ i Δ j ( y ) ) 〉 = PG ( W n - i ⊕ W n - i + 1 ⊕ ⋯ ⊕ W n ) ; (4) dim ( W i ) = n i 2 - n i - 1 · n i + 1 .