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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