The Geometry of the Classical Groups

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