scholarly journals The Valuations of the Near Polygon ${\Bbb G}_n$

10.37236/226 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Polar Space ◽  
Near Polygon ◽  
Dual Polar Space

We show that every valuation of the near $2n$-gon ${\Bbb G}_n$, $n \geq 2$, is induced by a unique classical valuation of the dual polar space $DH(2n-1,4)$ into which ${\Bbb G}_n$ is isometrically embeddable.

scholarly journals The Valuations of the Near Octagon ${\Bbb I}_4$

10.37236/1102 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Bart De Bruyn ◽  
Pieter Vandecasteele
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Near Polygon ◽  
Hyperbolic Quadric ◽  
Dual Polar Space ◽  
Full Embedding ◽  
Parabolic Quadric ◽  
Dual Polar Spaces

The maximal and next-to-maximal subspaces of a nonsingular parabolic quadric $Q(2n,2)$, $n \geq 2$, which are not contained in a given hyperbolic quadric $Q^+(2n-1,2) \subset Q(2n,2)$ define a sub near polygon ${\Bbb I}_n$ of the dual polar space $DQ(2n,2)$. It is known that every valuation of $DQ(2n,2)$ induces a valuation of ${\Bbb I}_n$. In this paper, we classify all valuations of the near octagon ${\Bbb I}_4$ and show that they are all induced by a valuation of $DQ(8,2)$. We use this classification to show that there exists up to isomorphism a unique isometric full embedding of ${\Bbb I}_n$ into each of the dual polar spaces $DQ(2n,2)$ and $DH(2n-1,4)$.

scholarly journals The Universal Embedding of the Near Polygon ${\Bbb G}_n$

10.37236/957 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Automorphism Group ◽  
Polar Space ◽  
Near Polygon ◽  
Dual Polar Space ◽  
Large Automorphism Group ◽  
Universal Embedding

In an earlier paper, we showed that the dual polar space $DH(2n-1,4)$, $n \geq 2$, has a sub near-$2n$-gon ${\Bbb G}_n$ with a large automorphism group. In this paper, we determine the absolutely universal embedding of this near polygon. We show that the generating and embedding ranks of ${\Bbb G}_n$ are equal to ${2n \choose n}$. We also show that the absolutely universal embedding of ${\Bbb G}_n$ is the unique full polarized embedding of this near polygon.

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

10.37236/90 ◽  
2009 ◽  
Vol 16 (1) ◽  
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.

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

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.

scholarly journals On the nucleus of the Grassmann embedding of the symplectic dual polar space DSp(2n,F), char(F)=2

2009 ◽  
Vol 30 (2) ◽  
pp. 468-472 ◽  
Author(s):  
Rieuwert J. Blok ◽  
Ilaria Cardinali ◽  
Bart De Bruyn
Keyword(s):  
Polar Space ◽  
Dual Polar Space ◽  
Symplectic Dual Polar Space

scholarly journals The non-existence of ovoids in the dual polar space DW(5,q)

2003 ◽  
Vol 104 (2) ◽  
pp. 351-364 ◽  
Author(s):  
Bruce N. Cooperstein ◽  
Antonio Pasini
Keyword(s):  
Polar Space ◽  
Dual Polar Space

scholarly journals The universal embedding dimension of the binary symplectic dual polar space

2003 ◽  
Vol 264 (1-3) ◽  
pp. 3-11 ◽  
Author(s):  
A. Blokhuis ◽  
A.E. Brouwer
Keyword(s):  
Polar Space ◽  
Embedding Dimension ◽  
Dual Polar Space ◽  
Universal Embedding ◽  
Symplectic Dual Polar Space

The ovoidal hyperplanes of a dual polar space of rank 4

2007 ◽  
Vol 7 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Harm Pralle ◽  
Sergey Shpectorov
Keyword(s):  
Polar Space ◽  
Dual Polar Space

scholarly journals A non-abelian representation of the dual polar space DQ(2n,2)

2009 ◽  
Vol 9 (1) ◽  
pp. 177-188 ◽  
Author(s):  
Kamal Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Polar Space ◽  
Dual Polar Space

scholarly journals On the Universal Embedding of the Sp2n(2) Dual Polar Space

2001 ◽  
Vol 94 (1) ◽  
pp. 100-117 ◽  
Author(s):  
Paul Li
Keyword(s):  
Polar Space ◽  
Dual Polar Space ◽  
Universal Embedding
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