On the Generation of Dual Polar Spaces of Symplectic Type over Finite Fields

✍ Scribed by B.N Cooperstein

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 326 KB
• Volume: 83
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1998.2875

✦ Synopsis

It is demonstrated that the dual polar space of type Sp 2n (q), q>2, can be generated as a geometry by ( 2n n )&( 2n n&2 ) points.

1998 Academic Press

1. Introduction

We assume the reader is familiar with the basic definitions relating to undirected graphs and linear incidence system or point-line geometry (as a standard reference see [2]). In particular: the distance function, a geodesic path, and diameter of a graph; the collinearity graph of a point-line geometry 1=(P, L), a subspace of 1, the subspace (X) 1 generated by a subset X of P, and convex subspace of 1. We define the generating rank, gr(1), of a point-line geometry 1 to be min[ |X |: X/P, (X) 1 =P], that is, the minimal cardinality of a generating set of 1.

Let G=(P, L) be a point-line geometry. By a projective embedding of 1 we mean an injective mapping e: P Ä PG(V ), V a vector space over some division ring, such that (i) the space spanned by e(P) is all of PG(V ) and (ii) for l # L, e(l ) is a full line of PG(V ). We say that 1 is embeddable if some projective embedding of 1 exists. When 1 is embeddable, we define the embedding rank, er(1 ), of 1 to the maximal dimension of a vector space V for which there exists an embedding into PG(V ). Suppose now that 1=(P, L) is a point-line geometry and e i : P Ä PG(V i ), i=1, 2 are projective embeddings. A morphism of embeddings is a map :: PG(V 1 ) Ä PG(V 2 ) induced by a surjective semi-linear transformation of the underlying vector spaces V 1 , V 2 such that : b e 1 =e 2 . An embedding e^is said to be universal relative to e if there