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

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 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 ei : P PG(Vi), i=1, 2 are projective embeddings. A morphism of embeddings is a map :: PG(V1) PG(V2) induced by a surjective semi-linear transformation of the underlying vector spaces V1 , V2 such that : b e1=e2 . An embedding ê is said to be universal relative to e if there is a morphism :̂: ê e such that for any other morphism #: e$ e, :̂ factors through #, that is, there is a morphism :$: ê e such that ê=:$ b #. An embedding e: P PG(W ) is relatively Article No. TA982875