Generating Symplectic and Hermitian Dual Polar Spaces over Arbitrary Fields Nonisomorphic to ${\Bbb F}_2$

Abstract

Cooperstein proved that every finite symplectic dual polar space $DW(2n-1,q)$, $q \neq 2$, can be generated by ${2n \choose n} - {2n \choose n-2}$ points and that every finite Hermitian dual polar space $DH(2n-1,q^2)$, $q \neq 2$, can be generated by ${2n \choose n}$ points. In the present paper, we show that these conclusions remain valid for symplectic and Hermitian dual polar spaces over infinite fields. A consequence of this is that every Grassmann-embedding of a symplectic or Hermitian dual polar space is absolutely universal if the (possibly infinite) underlying field has size at least 3.