On the nucleus of the Grassmann embedding of the symplectic dual polar space DSp(2n,F)DSp(2n,F).

Let \geq 3$and let$ be a field of characteristic 2. Let (2n,F)$denote the dual polar space associated with the building of Type$ over $and let$\mathcal{G}_{n-2}$denote the$(n-2) of type $. Using the bijective correspondence between the points of$\mathcal{G}_{n-2}$and the quads of (2n,F)$, we construct a full projective embedding of $\mathcal{G}_{n-2}$ into the nucleus of the Grassmann embedding of (2n,F)$. This generalizes a result of Cardinali and Lunardon which contains an alternative proof of this fact in the case when =3$ and $ is finite

A property of isometric mappings between dual polar spaces of type DQ(2n,K)

Let f be an isometric embedding of the dual polar space Delta = DQ(2n, K) into Delta' = DQ(2n, K'). Let P denote the point-set of Delta and let e' : Delta' -> Sigma' congruent to PG(2(n) - 1, K') denote the spin-embedding of Delta'. We show that for every locally singular hyperplane H of Delta, there exists a unique locally singular hyperplane H' of Delta' such that f(H) = f(P) boolean AND H'. We use this to show that there exists a subgeometry Sigma congruent to PG(2(n) - 1, K) of Sigma' such that: (i) e' circle f (x) is an element of Sigma for every point x of Delta; (ii) e := e' circle f defines a full embedding of Delta into Sigma, which is isomorphic to the spin-embedding of Delta

The hyperplanes of DQ(-)(7,K) arising from embedding

AbstractWe determine all hyperplanes of the dual polar space DQ−(7,K) which arise from embedding. This extends one of the results of [B. De Bruyn. The hyperplanes of DQ(2n,K) and DQ−(2n+1,q) which arise from their spin-embeddings, J. Combin. Theory Ser. A 114 (2007), 681–691] to the infinite case

A decomposition of the universal embedding space for the near polygon ℍn

Let H-n, n >= 1, be the near 2n-gon defined on the 1-factors of the complete graph on 2n + 2 vertices, and let e denote the absolutely universal embedding of Hn into PG(W), where W is a 1/n((2n +2)(n + 1))-dimensional vector space over the field F-2 with two elements. For every point z of H-n and every i is an element of N, let Delta(i) (z) denote the set of points of H-n at distance i from z. We show that for every pair {x, y} of mutually opposite points of H-n, W can be written as a direct sum W-0 circle plus W-1 ... circle plus W-n such that the following four properties hold for every i is an element of {0,..., n}: (1) = PG(W-i); (2) = PG(W-0 circle plus ... circle plus W-i); (3) = PG(Wn-i circle plus Wn-i+ 1 circle plus ... circle plus W-n); (4) dim(W-i) = |Delta(i) (x) boolean AND Delta(n-i)(y)| = ((n)(i)) - ((n)(i-1)).((n)(i+1))

Veronesean embeddings of dual polar spaces of orthogonal type

Given a point-line geometry P and a pappian projective space S,a veronesean embedding of P in S is an injective map e from the point-set of P to the set of points of S mapping the lines of P onto non-singular conics of S and such that e(P) spans S. In this paper we study veronesean embeddings of the dual polar space \Delta_n associated to a non-singular quadratic form q of Witt index n >= 2 in V = V(2n + 1; F). Three such embeddings are considered,namely the Grassmann embedding gr_n,the composition vs_n of the spin (projective) embedding of \Delta_n in PG(2n-1; F) with the quadric veronesean map of V(2n; F) and a third embedding w_n defined algebraically in the Weyl module V (2\lambda_n),where \lambda_n is the fundamental dominant weight associated to the n-th simple root of the root system of type Bn. We shall prove that w_n and vs_n are isomorphic. If char(F) is different from 2 then V (2\lambda_n) is irreducible and w_n is isomorphic to gr_n while if char(F) = 2 then gr_n is a proper quotient of w_n. In this paper we shall study some of these submodules. Finally we turn to universality,focusing on the case of n = 2. We prove that if F is a finite field of odd order q > 3 then sv_2 is relatively universal. On the contrary,if char(F) = 2 then vs_2 is not universal. We also prove that if F is a perfect field of characteristic 2 then vs_n is not universal,for any n>=2

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

Pseudo-embeddings of the (point, k-spaces)-geometry of PG(n, 2) and projective embeddings of DW(2n-1, 2)

We classify all homogeneous pseudo-embeddings of the point-line geometry defined by the points and k-dimensional subspaces of PG(n, 2), and use this to study the local structure of homogeneous full projective embeddings of the dual polar space DW(2n-1, 2). Our investigation allows us to distinguish n possible types for such homogeneous embeddings. For each of these n types, we construct a homogeneous full projective embedding of DW(2n - 1, 2)

The structure of the spin-embeddings of dual polar spaces and related geometries

AbstractIn [B. De Bruyn, A. Pasini, Minimal scattered sets and polarized embeddings of dual polar spaces, European J. Combin. 28 (2007) 1890–1909], it was shown that every full polarized embedding of a dual polar space of rank n≥2 has vector dimension at least 2n. In the present paper, we will give alternative proofs of that result which hold for more general classes of dense near polygons. These alternative proofs allow us to characterize full polarized embeddings of minimal vector dimension 2n. Using this characterization result, we can prove a decomposition theorem for the embedding space. We will use this decomposition theorem to get information on the structure of the spin-embedding of the dual polar space DQ(2n,K)