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Sigma Series in Pure Mathematics -- Volume 9

   Enlarged Picture

Donald E. Taylor

The Geometry of the Classical Groups

240 pages, hard cover, ISBN 3-88538-009-9, EUR 40.00, 1992

The author starts with the introduction of vector spaces, sesquilinear forms, and then studies the classical groups - special linear, symplectic, unitary and orthogonal groups - along the lines of E. Artin. Emphasis is placed on the "building" of the groups and their corresponding BN-pairs. Symplectic groups, unitary groups, orthogonal groups, and the Klein correspondance are thoroughly treated in individual chapters, each offering an abundance of exercises for deepening the understanding.

"It is therefore highly recommended to students beginning to work with classical groups and who want to get some knowledge about the interaction between groups, classical geometries, buildings, BN-pairs and modern treatments like diagram geometries. ... The book is carefully written. ... The book fills a gap in the existing literature." (G. Stroth, Zentralblatt f. Mathematik).

Zentralblatt-Review


Contents:

Preface v
     
  Chapter 1: Groups Acting on Sets 1
1.1 Exercises 4
     
  Chapter 2: Affine Geometry 6
2.1 Semilinear transformations 7
2.2 The affine group 9
2.3 Exercises 10
     
  Chapter 3: Projective Geometry 13
3.1 Axioms for projective geometry 15
3.2 Exercises 16
     
  Chapter 4: The General and Special Linear Groups 18
4.1 The dual space 18
4.2 The groups SL(V) and PSL(V) 19
4.3 Order formulae 19
4.4 The action of PSL(V) on P(V) 20
4.5 Transvections 20
4.6 The simplicity of PSL(V) 22
4.7 The groups PSL(2,q) 23
4.8 Exercises 25
     
  Chapter 5: BN-Pairs and Buildings 27
5.1 The BN-pair axioms 27
5.2 The Tits building 28
5.3 The BN-pair of SL(V) 28
5.4 Chambers 30
5.5 Flags and apartments 30
5.6 Panels 32
5.7 Split BN-pairs 33
5.8 Commutator relations 34
5.9 The Weyl group 35
5.10 Exercises 36
     
  Chapter 6: The 7-Point Plane and the group A7 40
6.1 The 7-point plane 40
6.2 The simple group of order 168 41
6.3 A geometry of 7-point planes 43
6.4 A geometry for A8 45
6.5 Exercises 46
     
  Chapter 7: Polar Geometry 50
7.1 The dual space 50
7.2 Correlations 51
7.3 Sesquilinear forms 52
7.4 Polarities 53
7.5 Quadratic forms 54
7.6 Witt's theorem 55
7.7 Bases of orthogonal hyperbolic pairs 59
7.8 The group ΓL*(V) 60
7.9 Flags and frames 61
7.10 The building of a polarity 63
7.11 Exercises 65
     
  Chapter 8: Symplectic Groups 68
8.1 Matrices 68
8.2 Symplectic Bases 69
8.3 Order formulae 70
8.4 The action of PSp(V) on P(V) 70
8.5 Symplectic transvections 71
8.6 The simplicity of PSp(V) 72
8.7 Symmetric groups 74
8.8 Symplectic BN-pairs 75
8.9 Symplectic Buildings 77
8.10 Exercises 78
     
  Chapter 9: BN-Pairs, Diagrams and Geometries 83
9.1 The BN-pair of a polar building 83
9.2 The Weyl group 87
9.3 Coxeter groups 90
9.4 The exchange condition 91
9.5 Reflections and the strong exchange condition 94
9.6 Parabolic subgroups of Coxeter groups 96
9.7 Complexes 97
9.8 Coxeter complexes 98
9.9 Buildings 99
9.10 Chamber systems 101
9.11 Diagram geometries 103
9.12 Abstract polar spaces 107
9.13 Exercises 108
     
  Chapter 10: Unitary Groups 114
10.1 Matrices 114
10.2 The field F 115
10.3 Hyperbolic pairs 116
10.4 Order formulae 117
10.5 Unitary transvections 118
10.6 Hyperbolic lines 119
10.7 The action of PSU(V) on isotropic points 120
10.8 Three-dimensional unitary groups 121
10.9 The group PSU(3,2) 123
10.10 The group SU(4,2) 125
10.11 The simplicity of PSU(V) 127
10.12 An example 130
10.13 Unitary BN-pairs 130
10.14 Exercises 131
     
  Chapter 11: Orthogonal Groups 136
11.1 Matrices 137
11.2 Finite Fields 138
11.3 Order formulae - one 140
11.4 Three-dimensional orthogonal groups 142
11.5 Degenerate polar forms and the group O(2m+1,2k) 143
11.6 Reflections 144
11.7 Root groups 146
11.8 Siegel transformations 148
11.9 The action of PΩ(V) on singular points 150
11.10 Wall's parametrization of O(V) 153
11.11 Factorization theorems 155
11.12 The generation of O(V) by reflections 156
11.13 Dickson's invariant 160
11.14 The simplicity of PΩ(V) 160
11.15 The spinor norm 163
11.16 Order formulae - two 165
11.17 The groups PΩ(2m+1,q), q odd 166
11.18 Orthogonal BN-pairs 168
11.19 Maximal totally singular subspaces 170
11.20 The oriflamme geometry 172
11.21 Exercises 174
     
  Chapter 12: The Klein Correspondance 179
12.1 The exterior algebra of a vector space 180
12.2 The dual space 183
12.3 Decomposable k-vectors 183
12.4 Creation and annihilation operators 184
12.5 The Klein quadric 187
12.6 The groups SL(V) and Ω(Λ2(V)) 190
12.7 Correlations 191
12.8 Alternating forms and reflections 195
12.9 Hermitian forms of Witt index 2 196
12.10 Four-dimensional orthogonal groups 199
12.11 Generalized quadrangles and duality 201
12.12 The Suzuki groups 202
12.13 Exercises 207
     
  Bibliography 213
  Index of Symbols 221
  Index of names 223
  Subject Index 225