Near Polygons Having a big sub Near Polygon Isomorphic to $ \mathbb{H}_n $

2002 ◽  
Vol 6 (3) ◽  
pp. 285-294 ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Near Polygon

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

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 $(4,1)$.

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 Hyperplanes of the M 24 Near Polygon

2002 ◽  
Vol 18 (3) ◽  
pp. 415-420 ◽  
Author(s):  
Andries E. Brouwer ◽  
Hans Cuypers ◽  
Ernst W. Lambeck
Keyword(s):  
Near Polygon

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 An Alternative Definition of the Notion Valuation in the Theory of Near Polygons

10.37236/105 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Polar Spaces ◽  
Alternative Definition ◽  
Near Polygon ◽  
Point Set ◽  
Definition Of ◽  
Dual Polar Spaces

Valuations of dense near polygons were introduced in [9]. A valuation of a dense near polygon ${\cal S}=({\cal P},{\cal L},{\rm I})$ is a map $f$ from the point-set ${\cal P}$ of ${\cal S}$ to the set $\Bbb N$ of nonnegative integers satisfying very nice properties with respect to the set of convex subspaces of ${\cal S}$. In the present paper, we give an alternative definition of the notion valuation and prove that both definitions are equivalent. In the case of dual polar spaces and many other known dense near polygons, this alternative definition can be significantly simplified.

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