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

Abstract 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).

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

The Electronic Journal of Combinatorics ◽

10.37236/90 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 4

Author(s):

Bart De Bruyn

Keyword(s):

Polar Space ◽

Polar Spaces ◽

Hermitian Variety ◽

Projective Embedding ◽

Dual Polar Space ◽

Transfinite Recursion ◽

Symplectic Dual Polar Space ◽

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.

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Characterizations of the $G_2(4)$ and $L_3(4)$ Near Octagons

The Electronic Journal of Combinatorics ◽

10.37236/8476 ◽

2021 ◽

Vol 28 (4) ◽

Author(s):

Bart De Bruyn

Keyword(s):

Generalized Quadrangle ◽

Line Geometry ◽

Near Polygon ◽

Generalized Polygon ◽

Generalized Hexagon ◽

Split Cayley Hexagon ◽

Point Line ◽

Line Spread

A triple $(\mathcal{S},S,\mathcal{Q})$ consisting of a near polygon $\mathcal{S}$, a line spread $S$ of $\mathcal{S}$ and a set $\mathcal{Q}$ of quads of $\mathcal{S}$ is called a polygonal triple if certain nice properties are satisfied, among which there is the requirement that the point-line geometry $\mathcal{S}'$ formed by the lines of $S$ and the quads of $\mathcal{Q}$ is itself also a near polygon. This paper addresses the problem of classifying all near polygons $\mathcal{S}$ that admit a polygonal triple $(\mathcal{S},S,\mathcal{Q})$ for which a given generalized polygon $\mathcal{S}'$ is the associated near polygon. We obtain several nonexistence results and show that the $G_2(4)$ and $L_3(4)$ near octagons are the unique near octagons that admit polygonal triples whose quads are isomorphic to the generalized quadrangle $W(2)$ and whose associated near polygons are respectively isomorphic to the dual split Cayley hexagon $H^D(4)$ and the unique generalized hexagon of order $(4,1)$.

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Laguerre geometries and some connections to generalized quadrangles

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700037964 ◽

2007 ◽

Vol 83 (3) ◽

pp. 335-356

Author(s):

Matthew R. Brown

Keyword(s):

Three Dimensional ◽

Line Pair ◽

Generalized Quadrangles ◽

Constant Number ◽

Unique Point ◽

Line Geometry ◽

Laguerre Geometry ◽

Point Line ◽

Dimensional Projective Space ◽

Unique Circle

AbstractA Laguerre plane is a geometry of points, lines and circles where three pairwise non-collinear points lie on a unique circle, any line and circle meet uniquely and finally, given a circle C and a point Q not on it for each point P on C there is a unique circle on Q and touching C at P. We generalise to a Laguerre geometry where three pairwise non-collinear points lie on a constant number of circles. Examples and conditions on the parameters of a Laguerre geometry are given.A generalized quadrangle (GQ) is a point, line geometry in which for a non-incident point, line pair (P. m) there exists a unique point on m collinear with P. In certain cases we construct a Laguerre geometry from a GQ and conversely. Using Laguerre geometries we show that a GQ of order (s. s2) satisfying Property (G) at a pair of points is equivalent to a configuration of ovoids in three-dimensional projective space.

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Non-Classical Hyperplanes of $DW(5,q)$

The Electronic Journal of Combinatorics ◽

10.37236/2425 ◽

2013 ◽

Vol 20 (2) ◽

Author(s):

Bart De Bruyn

Keyword(s):

Prime Number ◽

Polar Space ◽

Dual Polar Space ◽

Singular Hyperplane ◽

Symplectic Dual Polar Space

The hyperplanes of the symplectic dual polar space $DW(5,q)$ arising from embedding, the so-called classical hyperplanes of $DW(5,q)$, have been determined earlier in the literature. In the present paper, we classify non-classical hyperplanes of $DW(5,q)$. If $q$ is even, then we prove that every such hyperplane is the extension of a non-classical ovoid of a quad of $DW(5,q)$. If $q$ is odd, then we prove that every non-classical ovoid of $DW(5,q)$ is either a semi-singular hyperplane or the extension of a non-classical ovoid of a quad of $DW(5,q)$. If $DW(5,q)$, $q$ odd, has a semi-singular hyperplane, then $q$ is not a prime number.

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On point-line geometry and displacement

Mechanism and Machine Theory ◽

10.1016/j.mechmachtheory.2004.05.004 ◽

2004 ◽

Vol 39 (10) ◽

pp. 1033-1050 ◽

Cited By ~ 8

Author(s):

Yi Zhang ◽

Kwun-Lon Ting

Keyword(s):

Line Geometry ◽

Point Line

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Veldkamp spaces: From (Dynkin) diagrams to (Pauli) groups

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817500803 ◽

2017 ◽

Vol 14 (05) ◽

pp. 1750080

Author(s):

Metod Saniga ◽

Frédéric Holweck ◽

Petr Pracna

Keyword(s):

Incidence Structure ◽

Polar Space ◽

Exceptional Point ◽

Specific Point ◽

Fano Plane ◽

Magic Square ◽

Dynkin Diagrams ◽

Pauli Matrix ◽

Matrix Formula ◽

Point Line

Regarding a Dynkin diagram as a specific point-line incidence structure (where each line has just two points), one can associate with it a Veldkamp space. Focusing on extended Dynkin diagrams of type [Formula: see text], [Formula: see text], it is shown that the corresponding Veldkamp space always contains a distinguished copy of the projective space PG[Formula: see text]. Proper labeling of the vertices of the diagram (for [Formula: see text]) by particular elements of the two-qubit Pauli group establishes a bijection between the 15 elements of the group and the 15 points of the PG[Formula: see text]. The bijection is such that the product of three elements lying on the same line is the identity and one also readily singles out that particular copy of the symplectic polar space [Formula: see text] of the PG[Formula: see text] whose lines correspond to triples of mutually commuting elements of the group; in the latter case, in addition, we arrive at a unique copy of the Mermin–Peres magic square. In the case of [Formula: see text], a more natural labeling is that in terms of elements of the three-qubit Pauli group, furnishing a bijection between the 63 elements of the group and the 63 points of PG[Formula: see text], the latter being the maximum projective subspace of the corresponding Veldkamp space; here, the points of the distinguished PG[Formula: see text] are in a bijection with the elements of a two-qubit subgroup of the three-qubit Pauli group, yielding a three-qubit version of the Mermin–Peres square. Moreover, save for [Formula: see text], each Veldkamp space is also endowed with some exceptional point(s). Interestingly, two such points in the [Formula: see text] case define a unique Fano plane whose inherited three-qubit labels feature solely the Pauli matrix [Formula: see text].

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