Adv. Geom. 5 (2005), 15–36 Advances in Geometry ( de Gruyter 2005 Witt-type theorems for Grassmannians and Lie incidence geometries B. N. Cooperstein, Anna Kasikova and Ernest E. Shult (Communicated by A. Pasini) Abstract. For a subset of a Lie incidence geometry two intrinsic notions of independence are introduced. Also defined is the notion of a parabolic subspace. A classification is achieved for certain independent subgraphs of the point collinearity graph of the Lie incidence geometries A n; k ; B n; n ; D n; n . As a corollary it is proved that certain subspaces of these geometries are par- abolic and transitivity results are obtained. Key words. Incidence geometry, Lie incidence geometry, singular independence, local inde- pendence, parabolic subspace. 1 Introduction and basic concepts We assume the reader is familiar with the concepts of a partial linear rank two inci- dence geometry G ¼ðP; LÞ and the Lie incidence geometries. We refer to articles in [2]. The collinearity graph of G is the graph ðP; DÞ where D consists of all pairs of points belonging to a common line. For a point x A P we will denote by DðxÞ the collection of all points collinear with x. For points x; y A P and a positive integer d a path of length d from x to y is a sequence x 0 ¼ x; x 1 ; ... ; x d ¼ y such that fx i ; x iþ1 g A D for each i ¼ 0; 1; ... ; d  1. The distance from x to y, denoted by dðx; yÞ, is defined to be the length of a shortest path from x to y if some path exists and otherwise is þy. For a point x and positive integer d we will let D d ðxÞ denote all points at distance d from x and by D cd ðxÞ all points at distance at most d from x. Note that D 1 ðxÞ¼ DðxÞ. By a subspace of G we mean a subset S such that if l A L and l V S contains at least two points, then l H S. ðP; LÞ is said to be Gamma space if, for every x A P, fxg U DðxÞ is a subspace. A subspace S is singular provided each pair of points in S is collinear, that is, S is a clique in the collinearity graph of G. For a Lie incidence geometry with respect to a ‘‘good node’’ every singular subspace, together with the lines it contains, is isomorphic to a projective space, see [4]. Clearly the intersection of subspaces is a subspace and consequently it is natural to define the subspace gen- Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/16/15 9:02 PM