Generalized Quadrangles with a Spread of Symmetry

Experimental Mathematics

2015

Self Cite

For already many years now one of the most important open problems in the theory of generalized quadrangles is whether other classes of generalized quadrangles exist besides those that are currently known. This paper reports on an unsuccessful attempt to construct a new generalized quadrangle. As a by-product of our attempt, we however obtain the following new characterization result: every generalized quadrangle of order 5 that has at least one regular point is isomorphic to the quadrangle W (5) arising from a symplectic polarity of PG (3,5). During the classification process we used the computer algebra system GAP to do certain computations or to experiment about an optimal strategy for the proof.

Section: Admissible Triplesmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Every Generalized Quadrangle of Order 5 Having a Regular Point Is Symplectic

Experimental Mathematics

2015

Self Cite

“…As a consequence, there are no glued near polygons of type E 1 Wð2Þ. Also, by [8], there is, up to equivalence, a unique spread of symmetry in Qð5; 2Þ. The spreads of symmetry of Qð5; 2Þ dualize to ovoids of symmetry in the point-line dual Hð3; 4Þ of Qð5; 2Þ.…”

Section: Proof Of Theoremmentioning

confidence: 91%

“…By [8], the generalized quadrangle Wð2Þ has no spread of symmetry. As a consequence, there are no glued near polygons of type E 1 Wð2Þ.…”

Section: Proof Of Theoremmentioning

confidence: 99%

The valuations of the near hexagons related to the Witt designs S(5,6,12) and S(5,8,24)

J of Combinatorial Designs

Vandecasteele

2005

Self Cite

Valuations of dense near polygons were introduced in [16]. In the present paper, we classify all valuations of the near hexagons E 1 and E 2 , which are related to the respective Witt designs Sð5,6,12Þ and Sð5,8,24Þ. Using these classifications, we prove that if a dense near polygon S contains a hex H isomorphic to E 1 or E 2 , then H is classical in S. We will use this result to determine all dense near octagons that contain a hex isomorphic to E 1 or E 2 . As a by-product, we obtain a purely geometrical proof for the nonexistence of regular near 2d-gons, d ! 4, whose parameters s, t, t i (0 i d) satisfy ðs, t 2 , t 3 Þ ¼ ð2, 1, 11Þ or ð2, 2, 14Þ. The nonexistence of these regular near polygons can also be shown with the aid of eigenvalue techniques.

“…Admissible triples were introduced in [2] to coordinatize generalized quadrangles with a so-called spread of symmetry, see also Section 2.2. They can also be used to coordinatize glued near hexagons, see Section 2.3.…”

Section: Definitionmentioning

confidence: 99%

Valuations of glued near hexagons

J of Combinatorial Designs

2006

Self Cite

Valuations of near polygons were introduced in [12] as an important tool for classifying dense near polygons. In the present article, we will introduce the class of the semidiagonal valuations. These valuations live in glued near hexagons. A glued near hexagon S can be coordinatized by a pair of admissible triples; such triples consist of a Steiner system L, a group G, and a certain nice map Á : L Â L ! G. We will give a necessary and sufficient condition for the existence of semi-diagonal valuations in S in terms of these two admissible triples. For two classes of glued near hexagons, we will use this condition to determine all semidiagonal valuations. Each semi-diagonal valuation will also give rise to a hyperplane of the glued near hexagon. # 2006 Wiley Periodicals, Inc. J Combin Designs 15: 35-48, 2007