De Bruyn Near Polygons

2006
ISBN: 978-3-7643-7553-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 263 Seiten

Reihe: Frontiers in Mathematics

Dedicated to the Russian mathematician Albert Shiryaev on his 70th birthday, this is a collection of papers written by his former students, co-authors and colleagues. The book represents the state-of-the-art of a quickly maturing theory and will be an essential source for researchers in this area. The diversity of topics and comprehensive style of the papers make the book attractive for Ph.D. students and young researchers.

Autoren/Hrsg.

De Bruyn, Bart

Inhaltsverzeichnis

1;Title page ;4
2;Copyright page ;5
3;Table of contents ;6
4;Preface;9
5;Chapter 1 Introduction;12
5.1;1.1 Definition of near polygon;12
5.2;1.2 Genesis;13
5.3;1.3 Near polygons with an order;14
5.4;1.4 Parallel lines;14
5.5;1.5 Substructures;15
5.6;1.6 Product near polygons;18
5.7;1.7 Existence of quads;23
5.8;1.8 The point-quad and line-quad relations;25
5.9;1.9 Some classes of near polygons;28
5.9.1;1.9.1 Thin and slim near polygons;28
5.9.2;1.9.2 Dense near polygons;28
5.9.3;1.9.3 Regular near polygons;28
5.9.4;1.9.4 Generalized polygons;29
5.9.5;1.9.5 Dual polar spaces;30
5.10;1.10 Generalized quadrangles of order (2, t);32
5.10.1;1.10.1 Examples;32
5.10.2;1.10.2 Possible orders;33
5.10.3;1.10.3 Generalized quadrangles of order (2, 1);34
5.10.4;1.10.4 Generalized quadrangles of order 2;34
5.10.5;1.10.5 Generalized quadrangles of order (2, 4);34
5.10.6;1.10.6 Ovoids in generalized quadrangles of order (2, t);35
6;Chapter 2 Dense near polygons;37
6.1;2.1 Main results;37
6.2;2.2 The existence of convex subpolygons;38
6.3;2.3 Proof of Theorem 2.6;47
6.4;2.4 Upper bound for the diameter of Gd(x);48
6.5;2.5 Upper bounds for t + 1 in the case of slim densenear polygons;50
6.6;2.6 Slim dense near polygons with a big convexsubpolygon;51
6.6.1;2.6.1 Statement of the result;51
6.6.2;2.6.2 Proof of Theorem 2.40;52
7;Chapter 3 Regular near polygons;56
7.1;3.1 Introduction;56
7.2;3.2 Some restrictions on the parameters;56
7.3;3.3 Eigenvalues of the collinearity matrix;59
7.3.1;Calculation of the multiplicities;61
7.3.2;Example 1: The case of regular near hexagons;62
7.3.3;Example 2: The case of regular near octagons;63
7.4;3.4 Upper bounds for t;63
7.5;3.5 Slim dense regular near hexagons;64
7.6;3.6 Slim dense regular near octagons;65
8;Chapter 4 Glued near polygons;66
8.1;4.1 Characterizations of product near polygons;66
8.2;4.2 Admissible d-spreads;71
8.3;4.3 Construction and elementary properties of glued near polygons ;72
8.4;4.4 Basic characterization result for glued nearpolygons;77
8.5;4.5 Other characterizations of glued near polygons;80
8.5.1;4.5.1 Characterization of finite glued near hexagons;80
8.5.2;4.5.2 Characterization of general glued near polygons;82
8.5.3;4.5.3 Proof of Theorem 4.28;83
8.6;4.6 Subpolygons;84
8.7;4.7 Glued near polygons of type d . {0, 1};86
8.7.1;4.7.1 Glued near polygons of type 0;86
8.7.2;4.7.2 Spreads of symmetry;86
8.7.3;4.7.3 Glued near polygons of type 1;89
8.7.4;4.7.4 Admissible triples;90
8.7.5;4.7.5 The sets .0(A) and .1(A) for a dense near polygon A;93
8.7.6;4.7.6 Extensions of spreads and automorphisms;94
8.7.7;4.7.7 Compatible spreads of symmetry;97
8.7.8;4.7.8 Compatible spreads of symmetry in product and glued nearpolygons;98
8.7.9;4.7.9 Near polygons of type (F1 * F2) . F3;99
9;Chapter 5 Valuations;102
9.1;5.1 Nice near polygons;102
9.2;5.2 Valuations of nice near polygons;103
9.3;5.3 Characterizations of classical and ovoidal valuations;105
9.4;5.4 The partial linear space Gf;107
9.5;5.5 A property of valuations;107
9.6;5.6 Some classes of valuations;108
9.6.1;5.6.1 Hybrid valuations;108
9.6.2;5.6.2 Product valuations;109
9.6.3;5.6.3 Diagonal valuations;110
9.6.4;5.6.4 Semi-diagonal valuations;110
9.6.5;5.6.5 Distance-j-ovoidal valuations;114
9.6.6;5.6.6 Extended valuations;115
9.6.7;5.6.7 SDPS-valuations;117
9.7;5.7 Valuations of dense near hexagons;118
9.8;5.8 Proof of Theorem 5.29;120
9.9;5.9 Proof of Theorem 5.30;124
9.10;5.10 Proof of Theorem 5.31;124
9.11;5.11 Proof of Theorem 5.32;125
10;Chapter 6 The known slim dense nearpolygons;130
10.1;6.1 The classical near polygons DQ(2n, 2) and DH(2n 1, 4);130
10.2;6.2 The class Hn;136
10.3;6.3 The class Gn;138
10.3.1;6.3.1 Definition of Gn;138
10.3.2;6.3.2 Subpolygons of Gn;140
10.3.3;6.3.3 Lines and quads in Gn;142
10.3.4;6.3.4 Some properties of Gn;143
10.3.5;6.3.5 Determination of Aut(Gn), n = 3;144
10.3.6;6.3.6 Spreads in Gn;146
10.3.7;6.3.7 Valuations of G3;148
10.4;6.4 The class In;149
10.5;6.5 The near hexagon E1;152
10.5.1;6.5.1 Description of E1 in terms of the extended ternary Golaycode;153
10.5.2;6.5.2 Description of E1 in terms of the Coxeter cap;154
10.5.3;6.5.3 The valuations of E1;158
10.6;6.6 The near hexagon E2;161
10.6.1;6.6.1 Definition and properties of E2;161
10.6.2;6.6.2 The ovoids of E2;164
10.7;6.7 The near hexagon E3;168
10.8;6.8 The known slim dense near polygons;170
10.9;6.9 The elements of C3 and C4;171
10.9.1;6.9.1 Spreads of symmetry of Q(5, 2);171
10.9.2;6.9.2 Another model for Q(5, 2);171
10.9.3;6.9.3 The near polygons DH(2n-1, 4).Q(5, 2), Gn .Q(5, 2) and E1 . Q(5, 2);173
10.9.4;6.9.4 Spreads of symmetry of Q(5, 2) . Q(5, 2);174
10.9.5;6.9.5 Near polygons of type (Q(5, 2) . Q(5, 2)) . Q(5, 2);174
11;Chapter 7 Slim dense near hexagons;176
11.1;7.1 Introduction;176
11.2;7.2 Elementary properties of slim dense near hexagons;177
11.3;7.3 Case I: S is a regular near hexagon;179
11.4;7.4 Case II: S contains grid-quads and W(2)-quads butno Q(5, 2)-quads;180
11.4.1;7.4.1 There exists a big W(2)-quad;180
11.4.2;7.4.2 No W(2)-quad is big;181
11.5;7.5 Case III: S contains grid-quads and Q(5, 2)-quads but no W(2)-quads;184
11.6;7.6 Case IV: S contains W(2)-quads and Q(5, 2)-quad sbut no grid-quads;185
11.7;7.7 Case V: S contains grid-quads, W(2)-quads and Q(5, 2)-quads;186
11.8;7.8 Appendix;190
12;Chapter 8 Slim dense near polygons with a nice chain of convex subpolygons;195
12.1;8.1 Overview;195
12.2;8.2 Proof of Theorem 8.1;197
12.3;8.3 Proof of Theorem 8.2;198
12.4;8.4 Proof of Theorem 8.3;201
12.5;8.5 Proof of Theorem 8.4;205
12.6;8.6 Proof of Theorem 8.5;205
12.7;8.7 Proof of Theorem 8.6;212
12.8;8.8 Proof of Theorem 8.7;212
12.9;8.9 Proof of Theorem 8.8;213
12.10;8.10 Proof of Theorem 8.9;216
13;Chapter 9 Slim dense near octagons;218
13.1;9.1 Some properties of slim dense near octagons;218
13.2;9.2 Existence of big hexes;219
13.3;9.3 Classification of the near octagons;226
14;Chapter 10 Nondense slim near hexagons;232
14.1;10.1 A few lemmas;232
14.2;10.2 Slim near hexagons with special points;233
14.2.1;10.2.1 Special points;233
14.2.2;10.2.2 Slim near hexagons of type (III);234
14.2.3;10.2.3 Slim near hexagons of type (II);234
14.2.4;10.2.4 Slim near hexagons of type (I);235
14.3;10.3 Slim near hexagons without special points;235
14.3.1;10.3.1 Examples;235
14.3.2;10.3.2 Upper bounds for the number of lines through a point;237
14.4;10.4 Proof of Theorem 10.8;237
14.5;10.5 Proof of Theorem 10.9;240
14.6;10.6 Proof of Theorem 10.10;242
14.6.1;10.6.1 Upper bound for |G3(x*)|;242
14.6.2;10.6.2 Some classes of paths in G3(x*);243
14.6.3;10.6.3 Some inequalities involving the values N(a, l) and Nl;246
14.6.4;10.6.4 The proof of Theorem 10.10;249
14.7;10.7 Slim near hexagons with an order;250
15;Appendix A Dense near polygons of order;254
15.1;A.1 Generalized quadrangles of order (3, t);254
15.2;A.2 Dense near hexagons of order (3, t);255
15.3;A.3 Dense near octagons of order (3, t);256
15.4;A.4 Some properties of dense near 2d-gons of order (3, t);257
15.5;A.5 Dense near polygons of order (3, t) with a nice chain of convex subpolygons;258
16;Bibliography;259
17;Index;266

Leseproben

"Preface (p. ix-x)

In this book, we intend to give an extensive treatment of the basic theory of general near polygons. The subject of near polygons has been around for about 25 years now. Excellent handbooks have appeared on certain important subclasses of near polygons like generalized quadrangles ([82]) and generalized polygons ([100]), but no book has ever occurred dealing with the topic of general near polygons. Although generalized polygons and especially generalized quadrangles are indispensable to the study of near polygons, we do not aim at giving a profound study of these incidence structures.

In fact, this book can be seen as complementary to the two above-mentioned books. Although generalized quadrangles and generalized polygons were intensively studied since they were introduced by Tits in his celebrated paper on triality ([96]), the terminology near polygon ?rst occurred in a paper in 1980. In [91], Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line geometries which they called near polygons. In [91], also some very fundamental results regarding the geometric structure of near polygons were obtained, like the existence of quads, a result which was later generalized by Brouwer and Wilbrink [16] who showed that any dense near polygon has convex subpolygons of any feasible diameter.

The paper [16] gives for the ?rst time a profound study of dense near polygons. Other important papers on near polygons from the 1980s and the beginning of the 1990s deal with dual polar spaces, the classi?cation of regular near polygons in terms of their parameters and the classi?cation of the slim dense near hexagons. The subject of near polygons has regained interest in the last years. Important new contributions to the theory were the theory of glued near polygons, the theory of valuations and important breakthrough results regarding the classi?cation of dense near polygons with three and four points on every line.

These new contributions will be discussed extensively in this book. This book essentially consists of two main parts. In the ?rst part of the book, which consists of the ?rst ?ve chapters, we develop the basic theory of near polygons. In Chapters 2, 3 and 4, we study three classes of near polygons: the dense, the regular and the glued near polygons.

Our treatment of the dense and glued near polygons is rather complete. The treatment of the regular near polygons is concise and results are not always accompanied with proofs. More detailed information on regular near polygons can be found in the book Distanceregular graphs [13] by Brouwer, Cohen and Neumaier. In that book regular near polygons are considered as one of the main classes of distance-regular graphs. In Chapter 5, we discuss the notion of valuation of a near polygon which is a very important tool for classifying near polygons."