Generalized hexagons of order (t,t)

ABSTRACT. Let P be a finite classical polar space of rank r, r t> 2. An ovoid O of P is a pointset of P, which has exactly one point in common with every totally isotropic subspace of rank r. It is proved that the polar space W,(q) arising from a symplectic polarity of PG(n, q), n odd and n > 3, that the polar space Q(2n, q) arising from a non-singular quadric in PG(2n, q), n > 2 and q even, that the polar space Q-(2n + 1, q) arising from a non-singular elliptic quadric in PG(2n + 1, q), n > 1, and that the polar space H(n, q2) arising from a non-singular Hermitian variety in PG(n, q2), n even and n > 2, have no ovoids.Let S be a generalized hexagon of order n (>/1). If V is a pointset of order n3+ 1 of S, such that every two points are at distance 6, then V is called an ovoid of S. If H(q) is the classical generalized hexagon arising from G2(q), then it is proved that H(q) has an ovoid iff Q(6, q) has an ovoid. There follows that Q(6, q), q = 3 zh+a, has an ovoid, and that H(q), q even, has no ovoid.A regular system of order m on H(3, q2) is a subset K of the lineset of H(3, q2), such that through every point of H(3, q2) there are m (> 0) lines of K. B. Segre shows that, if K exists, then m = q + 1 or (q + 1)/2. If m = (q + 1)/2, K is called a hemisystem. The last part of the paper gives a very short proof of Segre's result. Finally it is shown how to construct the 4-(11,5, 1) design out of the hemisystem with 56 lines (q = 3). OvoiDs AND SPREADSLet P be a finite classical polar space of rank (or index) r, r >/2 [3]. An ovoid O of P is a pointset of P, which has exactly one point in common with every totally isotropic subspace of rank r. A spread S of P is a set of maximal totally isotropic subspaces, which constitutes a partition of the pointset.We shall use the following notation: W,(q)the polar space arising from a symplectic polarity of PG(n, q), n odd; Q(2n, q) the polar space arising from a non-singular quadric Q in PG(2n, q);Q+ (2n + 1, q) the polar space arising from a non-singular hyperbolic quadric Q + [7] in PG(2n + 1, q); Q-(2n + 1, q) the polar space arising from a non-singular elliptic quadric Q- [7] in PG(2n + 1, q); H(n, q2) the polar space arising from a non-singular Hermitian variety H [7] in PG(n, q2).The following results are known:(a) W,(q), n odd, has always a (regular) spread (the proof given in [16] for n = 5, extends to any odd n). W3(q) has an ovoid iff q is even [19]; every ovoid of W3(q), q even, is an ordinary ovoid of PG(3, q) and every ordinary ovoid of PG(3, q), q even, is an ovoid of some W3(q) [18].

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“…Moreover, for x,y~P, x #y, we have 2(x,y)~<4 iff x and y are on a line of the polar space Q(6, q) (see also [24]). …”

Section: The Polar Space H'(/7 Q2) /7 Evenmentioning

confidence: 96%

Ovoids and spreads of finite classical polar spaces

1981

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“…with the singular points of a 7-dimensional orthogonal geometry in such a way that blocks are identified with the hyperplanes x + of that space. By [Ya,Theorem 1.1], [Ro] or [CK, Appendix], it follows that H(q) g is isomorphic to H(q). However, H(q) is not self-dual since q is not a power of 3.…”

Section: Recovering the Generalized Hexagon From Its Designmentioning

confidence: 98%

Symmetric Designs from theG2 (q) Generalized Hexagons

Dempwolff¹,

Kantor²

2002

Journal of Combinatorial Theory, Series A

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“…The study of generalized polygons has recently attracted a great deal of attention. In addition to Tits' and Weiss' work on Moufang polygons and their generalizations (discussed by Weiss at the conference), there have been major I and elegantly geometric characterizations of the generalized quadrangles [34] and hexagons [40], [26] which arise from Chevalley groups. As yet, 2Fiq) octagons have not been characterized geometrically.…”

Section: • • •mentioning

confidence: 99%

Further problems concerning finite geometries and finite groups

Kantor¹

1981

Proceedings of Symposia in Pure Mathematics

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This note is intended as a very long footnote to the other finite geometry papers in this volume. Areas of overlap between finite geometry (or combinatorics) and finite groups will be listed which were either not mentioned, or were only briefly alluded to, at the conference. However, the descriptions and bibliography given here will also be brief: the goal is to indicate research directions, not to give a comprehensive survey.

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Generalized hexagons of order (t,t)

Cited by 19 publications

References 6 publications

Ovoids and spreads of finite classical polar spaces

Ovoids and spreads of finite classical polar spaces

Symmetric Designs from theG2 (q) Generalized Hexagons

Further problems concerning finite geometries and finite groups

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