Bart De Bruyn

Valuations and hyperplanes of dual polar spaces

Valuations were introduced in De Bruyn and Vandecasteele (Valuations of near polygons, preprint, 2004) as a very important tool for classifying near polygons. In the present paper we study valuations of dual polar spaces. We will introduce the class of the SDPS-valuations and characterize these valuations. We will show that a valuation of a finite thick dual polar space is the extension of an SDPS-valuation if and only if no induced hex valuation is ovoidal or semi-classical. Each SDPS-valuation will also give rise to a geometric hyperplane of the dual polar space.

The structure of full polarized embeddings of symplectic and Hermitian dual polar spaces

Abstract Let Δ be a thick dual polar space of rank n ≥ 2 and let e be a full polarized embedding of Δ into a projective space ∑. For every point x of Δ and every i ∈ {0, …, n}, let T i (x) denote the subspace of ∑ generated by all points e(y) with d(x, y) ≤ i. We show that T i (x) does not contain points e(z) with d(x, z) ≥ i + 1. We also show that there exists a well-defined map ei x from the set of (i – 1)-dimensional subspaces of the residue Res Δ(x) of Δ at the point x (which is a projective space of dimension n – 1) to the set of points of the quotient space T i (x)/T i –1 (x). In this paper we study the structure of the maps ei x and the subspaces Ti(x) for some particular full polarized embeddings of the symplectic and the Hermitian dual polar spaces. Our investigations allow us to answer some questions asked in the literature.

Generalized Quadrangles with a Spread of Symmetry

We present a common construction for some known infinite classes of generalized quadrangles. Whether this construction yields other (unknown) generalized quadrangles is an open problem. The class of generalized quadrangles obtained this way is characterized in two different ways. On the one hand, they are exactly the generalized quadrangles having a spread of symmetry. On the other hand, they can be characterized in terms of the group of projectivities with respect to a spread. We explore some properties of these generalized quadrangles. All these results can be applied to the theory of the glued near hexagons, a class of near hexagons introduced by the author in De Bruyn (1998) On near hexagons and spreads of generalized quadrangles, preprint.

On the Grassmann-embeddings of the hermitian dual polar spaces

Let H denote a hermitian variety of Witt-index n ≥ 2 in PG(2n–1, 𝕂) and let θ denote the associated involutory automorphism of 𝕂. The dual polar space Δ associated with H has a full projective embedding e (the so-called Grassmann-embedding) into PG(W), where W is a -dimensional vector space over the fix-field 𝕂0 of θ. The projective space PG(W) can be regarded as a Baer-subgeometry of PG(∧ n V), where V is a 2n-dimensional vector space over 𝕂, equipped with a hermitian form defining H. In this article, we determine the precise equations for the Baer-subgeometry PG(W) of PG(∧ n V) and give a proof for the fact that e is an embedding without the use of any group-theoretical considerations. Subsequently, we will prove a decomposition theorem for the embedding e in the same spirit as the decomposition theorem for the Grassmann-embeddings of the symplectic dual polar spaces [B. De Bruyn, A decomposition of the natural embedding spaces for the symplectic dual polar spaces. Linear Algebra and its Applications, in press].Let H denote a hermitian variety of Witt-index n ≥ 2 in PG(2n–1, 𝕂) and let θ denote the associated involutory automorphism of 𝕂. The dual polar space Δ associated with H has a full projective embedding e (the so-called Grassmann-embedding) into PG(W), where W is a -dimensional vector space over the fix-field 𝕂0 of θ. The projective space PG(W) can be regarded as a Baer-subgeometry of PG(∧ n V), where V is a 2n-dimensional vector space over 𝕂, equipped with a hermitian form defining H. In this article, we determine the precise equations for the Baer-subgeometry PG(W) of PG(∧ n V) and give a proof for the fact that e is an embedding without the use of any group-theoretical considerations. Subsequently, we will prove a decomposition theorem for the embedding e in the same spirit as the decomposition theorem for the Grassmann-embeddings of the symplectic dual polar spaces [B. De Bruyn, A decomposition of the natural embedding spaces for the symplectic dual polar spaces. Linear Algebra and its Applications, i...

The hyperplanes of DH(5, q 2)

Abstract We classify all hyperplanes of the dual polar space DH(5, q 2), q ≠ 2. We will show that there are five types of hyperplanes and that all of them arise from the universal embedding of DH(5, q 2). These five types of hyperplanes also exist in the dual polar space DH(5, 4), but they are not all the hyperplanes. In the forthcoming paper [De Bruyn B. and Pralle H.: The exceptional hyperplanes of DH(5,4). European J. Combin. to appear] we will show that DH(5, 4) has four extra types of hyperplanes. 2000 Mathematics Subject Classification: 51A50, 51A45, 51E24.

The hyperplanes of DQ(2n,K) and DQ-(2n+1,q) which arise from their spin-embeddings

We characterize the hyperplanes of the dual polar spaces DQ(2n,K) and DQ^-(2n+1,q) which arise from their respective spin-embeddings. The hyperplanes of DQ(2n,K) which arise from its spin-embedding are precisely the locally singular hyperplanes of DQ(2n,K). The hyperplanes of DQ^-(2n+1,q) which arise from its spin-embedding are precisely the hyperplanes H of DQ^-(2n+1,q) which satisfy the following property: if Q is an ovoidal quad, then Q@?H is a classical ovoid of Q.

On Near Hexagons and Spreads of Generalized Quadrangles

The glueing-construction described in this paper makes use of two generalized quadrangles with a spread in each of them and yields a partial linear space with special properties. We study the conditions under which glueing will give a near hexagon. These near hexagons satisfy the nice property that every two points at distance 2 are contained in a quad. We characterize the class of the “glued near hexagons” and give examples, some of which are new near hexagons.The glueing-construction described in this paper makes use of two generalized quadrangles with a spread in each of them and yields a partial linear space with special properties. We study the conditions under which glueing will give a near hexagon. These near hexagons satisfy the nice property that every two points at distance 2 are contained in a quad. We characterize the class of the “glued near hexagons” and give examples, some of which are new near hexagons.

The exceptional hyperplanes of DH(5,4)

In [B. De Bruyn, H. Pralle, The hyperplanes of DH(5,q^2), preprint 2005], we showed the existence of five isomorphism classes of hyperplanes in each dual polar space of type DH(5,q^2). We also proved there that every hyperplane of DH(5,q^2) belongs to one of these five classes if q 2. In the present paper, we classify all hyperplanes of DH(5,4) and find four additional isomorphism classes of hyperplanes.

The hyperplanes of DW(5, 2h) which arise from embedding

We show that there are six isomorphism classes of hyperplanes of the dual polar space @D=DW(5,2^h) which arise from the Grassmann-embedding. If h>=2, then these are all the hyperplanes of @D arising from an embedding. If h=1, then there are 6 extra classes of hyperplanes as has been shown by Pralle [H. Pralle, The hyperplanes of DW(5,2), Experiment. Math. 14 (2005) 373-384] with the aid of a computer. We will give a computer-free proof for this fact. The hyperplanes of DW(5,q), q odd, arising from an embedding will be classified in the forthcoming paper [B.N. Cooperstein, B. De Bruyn, Points and hyperplanes of the universal embedding space of the dual polar space DW(5,q),q odd, Michigan Math. J. (in press)].

The classification of the slim dense near octagons

We classify all dense near octagons with three points on each line.