Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Generating symplectic and Hermitian dual polar spaces over arbitrary fields nonisomorphic to F2. | Generating symplectic and Hermitian dual polar spaces over arbitrary fields nonisomorphic to F2 Bart De Bruyn Department of Pure Mathematics and Computer Algebra Ghent University Gent Belgium bdb@ and Antonio Pasini Dipartimento di Scienze Matematiche e Informatiche Università di Siena Siena Italy pasini@ Submitted Jan 30 2007 Accepted Jul 29 2007 Published Aug 4 2007 Mathematics Subject ClassiHcations 51A45 51A50 Abstract Cooperstein 6 7 proved that every Hnite symplectic dual polar space DW 2n 1 q q 2 can be generated by dD C2A points and that every Hnite Hermitian n xn 2Ỉ dual polar space DH 2n 1 q2 q 2 can be generated by 2 points. In the present paper we show that these conclusions remain valid for symplectic and Hermitian dual polar spaces over inHnite Helds. A consequence of this is that every Grassmann-embedding of a symplectic or Hermitian dual polar space is absolutely universal if the possibly inHnite underlying Held has size at least 3. 1 Introduction Let r P L I be a partial linear space . a rank 2 geometry with point-set P line-set L and incidence relation I c P X L for which every line is incident with at least two points and every two distinct points are incident with at most 1 line. A subspace of r is a set of points which contains all the points of a line as soon as it contains at least two points of it. If X is a nonempty set of points of r then X r denotes the smallest subspace of r Postdoctoral Fellow of the Research Foundation - Flanders THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R54 1 containing the set X. The minimal number gr r min XI X c P and X r Pg of points which are necessary to generate the whole point-set P is called the generating rank of r. A full embedding e of r into a projective space s is an injective mapping e from P to the point-set of s satisfying i e P s S ii e L e x I x 2 Lg is a line of s for every line L of r. The numbers dim s and dim s 1 are respectively called the projective dimension and
