Dual Polar Spaces Of Rank Research Articles

Non-Classical Hyperplanes of Finite Thick Dual Polar Spaces

We obtain a classification of the nonclassical hyperplanes of all finite thick dual polar spaces of rank at least 3 under the assumption that there are no ovoidal and semi-singular hex intersections. In view of the absence of known examples of ovoids and semi-singular hyperplanes in finite thick dual polar spaces of rank 3, the condition on the nonexistence of these hex intersections can be regarded as not very restrictive. As a corollary, we also obtain a classification of the nonclassical hyperplanes of $DW(2n-1,q)$, $q$ even. In particular, we obtain a complete classification of all nonclassical hyperplanes of $DW(2n-1,q)$ if $q \in \{ 8,32 \}$.

Projective embeddings of dual polar spaces of mixed type

The classification of polar spaces was completed by Tits in his monumental work [19] who showed as an intermediate step that almost all of them were embeddable in a projective space. The current paper is part of a project to classify all full projective embeddings of the duals of polar spaces. A complete classification is available for rank 2 by the work of Tits [19], Buekenhout–Lefèvre [4] and Dienst [13]. Recently, significant process was made in the rank 3 case by De Bruyn and Van Maldeghem [12], who showed, among other things, that the members of five families of dual polar spaces of rank 3 related to alternative division rings have full projective embeddings. For two of these families, the quaternionic dual polar spaces and the dual polar spaces of mixed type, it was not yet known whether full projective embeddings exist for rank at least four. In the present paper, we prove that any “mixed dual polar space” of rank at least 2 has a full projective embedding.

Hyperplanes of Hermitian dual polar spaces of rank 3 containing a quad

Let \({\mathbb F}\) and \({\mathbb F}'\) be two fields such that \({\mathbb F}'\) is a quadratic Galois extension of \({\mathbb F}\). If \(|{\mathbb F}| \ge 3\), then we provide sufficient conditions for a hyperplane of the Hermitian dual polar space \(DH(5,{\mathbb F}')\) to arise from the Grassmann embedding. We use this to give an alternative proof for the fact that all hyperplanes of \(DH(5,q^2)\), \(q \not = 2\), arise from the Grassmann embedding, and to show that every hyperplane of \(DH(5,{\mathbb F}')\) that contains a quad \(Q\) is either classical or the extension of a non-classical ovoid of \(Q\). We will also give a classification of the hyperplanes of \(DH(5,{\mathbb F}')\) that contain a quad and arise from the Grassmann embedding.

Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras

Abstract Suppose 𝕆 is an alternative division algebra that is quadratic over some subfield 𝕂 of its center Z ( 𝕆 ) $Z(\mathbb {O})$ . Then with ( 𝕆 , 𝕂 ) $(\mathbb {O},\mathbb {K})$ , there is associated a dual polar space. We provide an explicit representation of this dual polar space into a ( 6 n + 7 ) $(6n+7)$ -dimensional projective space over 𝕂, where n = dim 𝕂 ( 𝕆 ) $n = \dim _\mathbb {K}(\mathbb {O})$ . We prove that this embedding is the universal one, provided | 𝕂 | > 2 $\vert \mathbb {K}\vert >2$ . When 𝕆 is not an inseparable field extension of 𝕂, we show that this universal embedding is the unique polarized one. When 𝕆 is an inseparable field extension of 𝕂, then we determine the minimal full polarized embedding, and show that all homogeneous embeddings are either universal or minimal. We also provide explicit generators of the corresponding projective representations of the little projective group associated with the (dual) polar space.

On hyperbolic sets of maxes in dual polar spaces

Suppose Δ is a fully embeddable thick dual polar space of rank n≥3. It is known that a hyperplane H of Δ is classical if all its nontrivial intersections with quads are classical. In order to conclude that a hyperplane H is classical, it is perhaps not necessary to require in advance that all these intersections are classical. In fact, in this paper we show that for dual polar spaces admitting hyperbolic sets of maxes, the existence of certain classical quad–hyperplane intersections implies that other quad–hyperplane intersections need to be classical as well. We will also derive necessary and sufficient conditions for two disjoint maxes to be contained in a (necessarily unique) hyperbolic set of maxes. Dual polar spaces admitting hyperbolic sets of maxes include all members of a class of embeddable dual polar spaces related to quadratic alternative division algebras.

New flag-transitive geometries for the groups $$Sp_4(K)$$ S p 4 ( K ) and $$SU_5(K)$$ S U 5 ( K )

In Pralle and Shpectorov (Adv Geom 7(1):1–17, 2007) the class of ovoidal hyperplanes in dual polar spaces of rank 4 is described. In this paper we observe that by removing such a hyperplane and a related second hyperplane one obtains a nice geometry for the group stabilising the ovoidal hyperplane. We show that this group acts flag-transitively and that the geometry is simply connected.

On the grassmann module of symplectic dual polar spaces of rank 4 in characteristic 3

Let V be the Weyl module of dimension 2 n n − 2 n n − 2 for the symplectic group Sp ( 2 n , F ) whose highest weight is the n th fundamental dominant weight. The module V affords the grassmann embedding of the symplectic dual polar space D W ( 2 n − 1 , F ) , therefore V is also called the grassmann module for the symplectic group. We consider the smallest case for char ( F ) odd for which V is reducible, namely n = 4 and char ( F ) = 3 . In this case the unique factor R of V has vector dimension 1. Here we provide a geometric description for R and study some relations between R and other objects associated with the grassmann embedding.

On a Class of Hyperplanes of the Symplectic and Hermitian Dual Polar Spaces

Let $\Delta$ be a symplectic dual polar space $DW(2n-1,{\Bbb K})$ or a Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$, $n \geq 2$. We define a class of hyperplanes of $\Delta$ arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$ arising from its Grassmann-embedding if and only if there exists an empty $\theta$-Hermitian variety in ${\rm PG}(n-1,{\Bbb K})$. Using this result we are able to give the first examples of ovoids in thick dual polar spaces of rank at least 3 which arise from some projective embedding. These are also the first examples of ovoids in thick dual polar spaces of rank at least 3 for which the construction does not make use of transfinite recursion.

On the nucleus of the Grassmann embedding of the symplectic dual polar space [formula omitted], [formula omitted

Let n ≥ 3 and let F be a field of characteristic 2. Let DSp ( 2 n , F ) denote the dual polar space associated with the building of type C n over F and let G n − 2 denote the ( n − 2 ) -Grassmannian of type C n . Using the bijective correspondence between the points of G n − 2 and the quads of DSp ( 2 n , F ) , we construct a full projective embedding of G n − 2 into the nucleus of the Grassmann embedding of DSp ( 2 n , F ) . This generalizes a result of an earlier paper [I. Cardinali, G. Lunardon, A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3, J. Combin. Theory Ser. A (in press)] which contains an alternative proof of this fact in the case when n = 3 and F is finite.

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.

On the simple connectedness of hyperplane complements in dual polar spaces

Let Δ be a dual polar space of rank n ≥ 4 , H be a hyperplane of Δ and Γ ≔ Δ ∖ H be the complement of H in Δ . We shall prove that, if all lines of Δ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings.

A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3

We give a geometrical description of the spin-embedding e sp of the symplectic dual polar space Δ ≅ DW ( 5 , 2 r ) by showing how the natural embedding of W ( 5 , 2 r ) into PG ( 5 , 2 r ) is involved in the Grassmann-embedding e gr of Δ. We prove that the map sending every quad of Δ to its nucleus realizes the natural embedding of W ( 5 , 2 r ) . Taking the quotient of e gr over the space spanned by the nuclei of the quadrics corresponding to the quads of Δ gives an embedding isomorphic to e sp .

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

In [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 2 n . 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 2 n . 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 ( 2 n , K ) .

The simple connectedness of hyperplane complements in thick dual polar spaces of rank at least 4

Abstract Let Δ be a dual polar space of rank n ⩾ 4 , H be a hyperplane of Δ and Γ : = Δ H be the complement of H in Δ. We shall prove that, if all lines of Δ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings.

Minimal full polarized embeddings of dual polar spaces

Let Δ be a thick dual polar space of rank n ≥ 2 admitting a full polarized embedding e in a finite-dimensional projective space Σ, i.e., for every point x of Δ, e maps the set of points of Δ at non-maximal distance from x into a hyperplane e∗(x) of Σ. Using a result of Kasikova and Shult [11], we are able the show that there exists up to isomorphisms a unique full polarized embedding of Δ of minimal dimension. We also show that e∗ realizes a full polarized embedding of Δ into a subspace of the dual of Σ, and that e∗ is isomorphic to the minimal full polarized embedding of Δ. In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces DQ(2n,q), DQ −(2n+1,q), DH(2n−1,q 2) and DW(2n−1,q) (q odd), but the latter only for n≤ 5. We shall prove that the minimal full polarized embeddings of DQ(2n,q), DQ −(2n+1,q) and DH(2n−1,q 2) are the `natural' ones, whereas this is not always the case for DW(2n−1, q).

Locally singular hyperplanes in thick dual polar spaces of rank 4

We study ( i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ ( 2 n , K ) , then we will show that every locally singular hyperplane of Δ is singular. We will describe a new type of hyperplane in DQ ( 8 , K ) and show that every locally singular hyperplane of DQ ( 8 , K ) is either singular, the extension of a hexagonal hyperplane in a hex or of the new type.

Hyperplanes of dual polar spaces of rank 3 with no subquadrangular quad

Hyperplanes of dual polar spaces of rank 3 with no subquadrangular quad

Point-line characterizations of Lie geometries

There are two basic theorems. Let G be a strong parapolar space with these three properties: (1) For each point x and symplecton S, x is collinear to some point of S. (2) The set of points at distance at most two from a point forms a geometric hyperplane. (3) If every sym- plecton has rank at least three, every maximal singular subspace has finite projective rank. Then G is either D6; 6; A5; 3 or E7; 1, a classical dual polar space of rank three, or a product geometry LP where P is a polar space and L is a line. The second theorem concerns parapolar spaces S of symplectic rank at least three whose point-collinearity diameter is at least three such that for every point-symplecton pair, ðx; SÞ, x ? V S is never just a point. With a mild local condi- tion, one can show that such a geometry has point-diameter three and has a simply connected point-collinearity graph. If singular spaces have finite projective rank, one can show that S is E6; 4, E7; 7, E8; 1, a metasymplectic space, or a polar Grassmannian of type Bn; 2, Dn; 2, nd 4. All of these geometries are truncations of buildings. The last case can be modified so that the as- sumption that singular spaces have finite projective rank can be discarded.

Uniform Hyperplanes of Finite Dual Polar Spaces of Rank 3

Let Δ be a finite thick dual polar space of rank 3. We say that a hyperplane H of Δ is locally singular (respectively, quadrangular or ovoidal) if H∩Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of Δ. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally singular, then either H is the set of points at non-maximal distance from a given point of Δ or Δ is the dual of Q(6, q) and H arises from the generalized hexagon H(q). In this paper we prove that only two examples exist for the locally quadrangular case, arising in Q(6, 2) and H(5, 4), respectively. We fail to rule out the locally ovoidal case, but we obtain some partial results on it, which imply that, in this case, the geometry Δ\\H induced by Δ on the complement of H cannot be flag-transitive. As a bi-product, the hyperplanes H with Δ\\H flag-transitive are classified.