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Báo cáo toán học: "Generating symplectic and Hermitian dual polar spaces over arbitrary fields nonisomorphic to F2"

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@cage.ugent.be and Antonio Pasini Dipartimento di Scienze Matematiche e Informatiche Università di Siena Siena Italy pasini@unisi.it 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 i.e. 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