17 research outputs found
Singer quadrangles
[no abstract available
An outline of polar spaces: basics and advances
This paper is an extended version of a series of lectures on polar spaces
given during the workshop and conference 'Groups and Geometries', held at the
Indian Statistical Institute in Bangalore in December 2012. The aim of this
paper is to give an overview of the theory of polar spaces focusing on some
research topics related to polar spaces. We survey the fundamental results
about polar spaces starting from classical polar spaces. Then we introduce and
report on the state of the art on the following research topics: polar spaces
of infinite rank, embedding polar spaces in groups and projective embeddings of
dual polar spaces
M-systems of polar spaces
AbstractLet P be a finite classical polar space of rank r, with r ⩾ 2. A partial m-system M of P, with 0 ≤ m ≦ r − 1, is any set {π1, π2, …, πk} of k (≠0) totally singular m-spaces of P such that no maximal totally singular space containing πi has a point in common with (ν1 ∪ π2 ∪ ⋯ ∪ πk) − πi, i = 1, 2, …, k. In each of the respective cases an upper bound δ for |M| is obtained. If |M| = δ, then M is called an m-system of P. For m = 0 the m-systems are the ovoids of P; for m = r − 1 the m-systems are the spreads of P. Surprisingly δ is independent of m, giving the explanation why an ovoid and a spread of a polar space P have the same size. In the paper many properties of m-systems are proved. We show that with m-systems of three types of polar spaces there correspond strongly regular graphs and two-weight codes. Also, we describe several ways to construct an m′-system from a given m-system. Finally, examples of m-systems are given