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 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

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 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

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 $DH(2n-1,4)$.

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 On the Universal Embedding of the U2n(2) Dual Polar Space

2002 ◽  
Vol 98 (2) ◽  
pp. 235-252 ◽  
Author(s):  
Paul Li
Keyword(s):  
Polar Space ◽  
Dual Polar Space ◽  
Universal Embedding

scholarly journals On sets of odd type of PG(n,4) and the universal embedding of the dual polar space DH(2n−1,4)

2012 ◽  
Vol 312 (3) ◽  
pp. 554-560 ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Polar Space ◽  
Dual Polar Space ◽  
Universal Embedding

scholarly journals Points and hyperplanes of the universal embedding space of the dual polar space DW (5, q ), q odd

2009 ◽  
Vol 58 (1) ◽  
pp. 195-212 ◽  
Author(s):  
Bruce Cooperstein ◽  
B. De Bruyn
Keyword(s):  
Polar Space ◽  
Dual Polar Space ◽  
Universal Embedding

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 Nearly all subspaces of a classical polar space arise from its universal embedding

2021 ◽  
Author(s):  
I. Cardinali ◽  
L. Giuzzi ◽  
A. Pasini
Keyword(s):  
Polar Space ◽  
Universal Embedding
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