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1.
2.
There is a new method of constructing generalized quadrangles (GQs) given by S. Löwe, which is based on covering of nets; all GQs with a regular point can be represented in this way. Here we give a method of constructing GQs with a regular point using the so-called content functions on nets. In the last part of the paper we lay the foundations for a research project aiming to use the more general notion of content to classify GQs and maybe to construct new ones.Both authors acknowledge the financial support by CRUI and DAAD in the frame of Programma Vigoni, which made this work possible. 相似文献
3.
《组合设计杂志》2018,26(7):356-366
Exploring the classical Ceva configuration in a Desarguesian projective plane, we construct two families of minimal blocking sets as well as a new family of blocking semiovals in PG(2, 32h). Also, we show that these blocking sets of PG(2, q2), regarded as pointsets of the derived André plane , are still minimal blocking sets in . Furthermore, we prove that the new family of blocking semiovals in PG(2, 32h) gives rise to a family of blocking semiovals in the André plane as well. 相似文献
4.
Peter Abramenko 《Indagationes Mathematicae》2004,15(3):305-319
Let Δ and Δ′ be two buildings of the same type (W, S), viewed as sets of chambers endowed with“distance” functions δ and δ′, respectively, admitting values in the common Weyl group W, which is a Coxeter group with standard generating set S. For a given element ω ε W, we study surjective maps ? : Δ → Δ′ with the property that δ(C, D) = ω if and only if Δ′ (?(C), ?(D)) = ω. The result is that the restrictions of ? to all residues of certain spherical types—determined by ω—are isomorphisms. We show with counterexamples that this result is optimal. We also demonstrate that, in many cases, this is enough to conclude that ? is an isomorphism. In particular, ? is an isomorphism if Δ and Δ′ are 2-spherical and every reduced expression of ω involves all elements of S. 相似文献
5.
This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s
2), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3
e
),e3, is constructed. Then, by the duality betweenQ(4, 3
e
) and the classical generalized quadrangleW (3
e
), we get line spreads of PG(3, 3
e
) and hence translation planes of order 32e
. These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q
2,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q
2),q even, having a subquadrangleS isomorphic toQ(4,q) and if inS each ovoid consisting of all points collinear with a given pointx ofS\S is an elliptic quadric, thenS is isomorphic toQ(5,q). 相似文献
6.
A concrete lattice chain model for the buildings of the classical groups over non archimedean complete skew fields is given.
The building axioms are proved in a uniform way using hereditary orders with involution instead of lattice chains.
Received: 16 December 1999 / Revised version: 12 July 2001 / Published online: 18 January 2002 相似文献
7.
Esmeralda L. N?stase 《Linear algebra and its applications》2011,435(6):1213-1221
A subspace partition of P=PG(n,q) is a collection of subspaces of P whose pairwise intersection is empty. Let σq(n,t) denote the minimum size (i.e., minimum number of subspaces) in a subspace partition of P in which the largest subspace has dimension t. In this paper, we determine the value of σq(n,t) for . Moreover, we use the value of σq(2t+2,t) to find the minimum size of a maximal partial t-spread in PG(3t+2,q). 相似文献
8.
A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that such an isomorphism between two such Kac-Moody groups induces an isomorphism between the respective twin root data of these groups. As a consequence, we obtain the solution of the isomorphism problem for Kac-Moody groups over finite fields of cardinality at least 4. 相似文献
9.
We prove that whenever a Kac-Moody group over a finite field is a lattice of its buildings, it has a fundamental domain with respect to which the induction cocycle is Lp for any p ∈ [1;+∞). The proof uses elementary counting arguments for root group actions on buildings. The applications are the possibility to apply some lattice superrigidity, and the normal subgroup property for Kac-Moody lattices.Prépublication de l’Institut Fourier nº 637 (2004); e-mail: http://www-fourier.ujf-grenoble.fr/prepublicatons.html 相似文献
10.
Stefan Immervoll 《Archiv der Mathematik》2006,87(5):478-480
We generalize a result of Kramer, see [7, 10.7 and 10.10], on generalized quadrangles associated with isoparametric hypersurfaces
of Clifford type to Tits buildings of type C2 derived from arbitrary isoparametric hypersurfaces with four distinct principal curvatures in spheres: two distinct points
p and q of a generalized quadrangle associated with an isoparametric hypersurface in the unit sphere of a Euclidean vector space
can be joined by a line K if and only if (p − q)/||p − q|| is a line. This line is orthogonal to K. Dually, two distinct lines L and K intersect if and only if (L − K)/||L − K|| is point.
Received: 14 October 2005 相似文献
11.
We transfer the whole geometry of PG(3, q) over a non-singular quadric Q4,q of PG(4, q) mapping suitably PG(3, q) over Q4,q. More precisely the points of PG(3, q) are the lines of Q4,q; the lines of PG(3, q) are the tangent cones of Q4,q and the reguli of the hyperbolic quadrics hyperplane section of Q4,q. A plane of PG(3, q) is the set of lines of Q4,q meeting a fixed line of Q4,q. We remark that this representation is valid also for a projective space
over any field K and we apply the above representation to construct maximal partial spreads
in PG(3, q). For q even we get new cardinalities for
For q odd the cardinalities are partially known. 相似文献
12.
A twin building is a pair of buildings along with a certain codistance' between chambers of one building, and chambers of the other. All spherical buildings can be regarded as twin buildings (such a building is twinned with itself in a natural way), and the aim of this paper is to give a definition of twin buildings along the lines that Tits originally gave for spherical buildings. This alternative approach via apartments has been used by the first author in studying some group actions on twin buildings. 相似文献
13.
Leanne D. Holder 《Journal of Geometry》2004,80(1-2):95-105
Define a conic blocking set to be a set of lines in a Desarguesian projective plane such that all conics meet these lines. Conic blocking sets can be used in determining if a collection of planes in projective three-space forms a flock of a quadratic cone. We discuss trivial conic blocking sets and conic blocking sets in planes of small order. We provide a construction for conic blocking sets in planes of non-prime order, and we make additional comments about the structure of these conic blocking sets in certain planes of even order. 相似文献
14.
In this paper, we first introduce new objects called “translation generalized ovals” and “translation generalized ovoids”,
and make a thorough study of these objects. We then obtain numerous new characterizations of the
of Tits and the classical generalized quadrangle
in even characteristic, including the complete classification of 2-transitive generalized ovals for the even case. Next,
we prove a new strong characterization theorem for the
of Tits. As a corollary, we obtain a purely geometric proof of a theorem of Johnson on semifield flocks.
* The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium). 相似文献
15.
Guglielmo Lunardon 《Combinatorica》2001,21(4):571-581
We point out the relationship between normal spreads and the linear k-blocking sets introduced in [9]. We give a characterisation of linear k-blocking sets proving that the projections and the embeddings of a PG(kt,q) in are linear k-blocking sets of . Finally, we construct some new examples.
Received December 19, 1997/Revised September 19, 2000
RID="*"
ID="*" Partially supported by Italian M.U.R.S.T. 相似文献
16.
The translation planes of order 81 admitting SL(2, 5), generated by affine elations, are completely determined. There are seven mutually non-isomorphic translation planes,
of which five are new. Each of these planes may be derived producing another set of seven mutually non-isomorphic translation
planes admitting SL(2, 5), where the 3-elements are Baer. Of this latter set, five planes are new. 相似文献
17.
Kac–Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices
in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but
not the only) examples of twin building lattices. We prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type (i.e. they
are not Bruhat–Tits buildings). Many of them are finitely presented and enjoy property (T). Our arguments explain geometrically
why simplicity fails to hold only for affine Kac–Moody groups. Moreover we prove that a nontrivial continuous homomorphism
from a completed Kac–Moody group is always proper. We also show that Kac–Moody lattices fulfill conditions implying strong
superrigidity properties for isometric actions on non-positively curved metric spaces. Most results apply to the general class
of twin building lattices.
Dedicated to Jacques Tits with our admiration 相似文献
18.
Alexander Kreuzer 《Geometriae Dedicata》1994,53(2):163-186
It is well known that every locally projective linear space (M,M) with dimM 3, fulfilling the Bundle Theorem (B) can be embedded in a projective space. We give here a new construction for the projective embedding of linear spaces which need not be locally projective. Essentially for this new construction are the assumptions (A) and (C) that for any two bundles there are two points on every line which are incident with a line of each of these bundles. With the Embedding Theorem (7.4) of this note for example a [0,m]-space can be embedded in a projective space. 相似文献
19.
We show that, if a collineation group G of a
generalized (2n + 1)-gon $\Gamma$ has the property that every symmetry of any
apartment extends uniquely to a collineation in G,
then $\Gamma$ is the unique projective plane with 3 points per line (the Fano plane)
and G is its full collineation group. A similar
result holds if one substitutes apartment with path of length
2k 2n + 2.Received: 19 June 2002 相似文献
20.
We give a characterization of the Buekenhout-Metz unitals in PG(2, q
2), in the cases that q is even or q=3, in terms of the secant lines through a single point of the unital. With the addition of extra conditions, we obtain further characterizations of Buekenhout-Metz unitals in PG(2, q
2), for all q. As an application, we show that the dual of a Buekenhout-Metz unital in PG(2, q
2) is a Buekenhout-Metz unital. 相似文献