Regular partitions of half-spin geometries

We describe several families of regular partitions of half-spin geometries and determine their associated parameters and eigenvalues. We also give a general method for computing the eigenvalues of regular partitions of half-spin geometries

Spin-embeddings, two-intersection sets and two-weight codes

Let Delta be one of the dual polar spaces DQ(8, q), DQ(-) (7, q), and let e : Delta -> Sigma denote the spin-embedding of Delta. We show that e(Delta) is a two-intersection set of the projective space Sigma. Moreover, if Delta congruent to DQ(-) (7, q), then e(Delta) is a (q(3) + 1)-tight set of a nonsingular hyperbolic quadric Q(+) (7, q(2)) of Sigma congruent to PG(7, q(2)). This (q(3) + 1)-tight set gives rise to more examples of (q(3) + 1)-tight sets of hyperbolic quadrics by a procedure called field-reduction. All the above examples of two-intersection sets and (q(3) + 1)-tight sets give rise to two-weight codes and strongly regular graphs

On geometric SDPS-sets of elliptic dual polar spaces

AbstractLet n∈N∖{0,1} and let K and K′ be fields such that K′ is a quadratic Galois extension of K. Let Q−(2n+1,K) be a nonsingular quadric of Witt index n in PG(2n+1,K) whose associated quadratic form defines a nonsingular quadric Q+(2n+1,K′) of Witt index n+1 in PG(2n+1,K′). For even n, we define a class of SDPS-sets of the dual polar space DQ−(2n+1,K) associated to Q−(2n+1,K), and call its members geometric SDPS-sets. We show that geometric SDPS-sets of DQ−(2n+1,K) are unique up to isomorphism and that they all arise from the spin embedding of DQ−(2n+1,K). We will use geometric SDPS-sets to describe the structure of the natural embedding of DQ−(2n+1,K) into one of the half-spin geometries for Q+(2n+1,K′)

Dual embeddings of dense near polygons

Let e: S -> Sigma be a full polarized projective embedding of a dense near polygon S, i.e., for every point p of S, the set H(p) of points at non-maximal distance from p is mapped by e into a hyperplane Pi(p) of Sigma. We show that if every line of S is incident with precisely three points or if S satisfies a certain property (P(de)) then the map p bar right arrow Pi p defines a full polarized embedding e* (the so-called dual embedding of e) of S into a subspace of the dual Sigma* of Sigma. This generalizes a result of [6] where it was shown that every embedding of a thick dual polar space has a dual embedding. We determine which known dense near polygons satisfy property (P(de)). This allows us to conclude that every full polarized embedding of a known dense near polygon has a dual embedding

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

Polarized non-abelian representations of slim near-polar spaces

In (Bull Belg Math Soc Simon Stevin 4:299-316, 1997), Shult introduced a class of parapolar spaces, the so-called near-polar spaces. We introduce here the notion of a polarized non-abelian representation of a slim near-polar space, that is, a near-polar space in which every line is incident with precisely three points. For such a polarized non-abelian representation, we study the structure of the corresponding representation group, enabling us to generalize several of the results obtained in Sahoo and Sastry (J Algebraic Comb 29:195-213, 2009) for non-abelian representations of slim dense near hexagons. We show that with every polarized non-abelian representation of a slim near-polar space, there is an associated polarized projective embedding