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

The hyperplanes of DW(5,F) arising from the Grassmann embedding

2017 ◽  
Vol 32 ◽  
pp. 1-14 ◽  
Author(s):  
Bart De Bruyn ◽  
Mariusz Kwiatkowski
Keyword(s):  
Main Tool ◽  
Complete Classification ◽  
Polar Space ◽  
Equivalence Classes ◽  
Dual Polar Space ◽  
Characteristic 2 ◽  
Universal Embedding ◽  
Symplectic Vector Space ◽  
Dimensional Module

The hyperplanes of the symplectic dual polar space DW(5; F) that arise from the Grassmann embedding have been classied in [B.N. Cooperstein and B. De Bruyn. Points and hyperplanes of the universal embedding space of the dual polar space DW(5; q), q odd. Michigan Math. J., 58:195{212, 2009.] in case F is a finite field of odd characteristic, and in [B. De Bruyn. Hyperplanes of DW(5;K) with K a perfect eld of characteristic 2. J. Algebraic Combin., 30:567{584, 2009.] in case F is a perfect eld of characteristic 2. In the present paper, these classifications are extended to arbitrary fields. In the case of characteristic 2 however, it was not possible to provide a complete classification. The main tool in the proof is the classification of the quasi-Sp(V; f)-equivalence classes of trivectors of a 6-dimensional symplectic vector space (V; f) obtained in [B. De Bruyn and M. Kwiatkowski. A 14-dimensional module for the symplectic group: orbits on vectors. Comm. Algebra,43:4553{4569, 2015.

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 Calculating the Dimension of the Universal Embedding of the Symplectic Dual Polar Space using Languages

10.37236/9754 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Carlos Segovia ◽  
Monika Winklmeier
Keyword(s):  
Polar Space ◽  
Symplectic Space ◽  
Alternative Proof ◽  
Isotropic Subspace ◽  
Dual Polar Space ◽  
Standard Order ◽  
Isotropic Subspaces ◽  
Universal Embedding ◽  
Totally Isotropic Subspaces ◽  
Symplectic Dual Polar Space

The main result of this paper is the construction of a bijection of the set of words in so-called standard order of length $n$ formed by four different letters and the set $\mathcal{N}^n$ of all subspaces of a fixed $n$-dimensional maximal isotropic subspace of the $2n$-dimensional symplectic space $V$ over $\mathbb{F}_2$ which are not maximal in a certain sense. Since the number of different words in standard order is known, this gives an alternative proof for the formula of the dimension of the universal embedding of a symplectic dual polar space $\mathcal{G}_n$. Along the way, we give formulas for the number of all $n$- and $(n-1)$-dimensional totally isotropic subspaces of $V$.

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.

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