Grassmannians of classical buildings

Contents......Page 10
Preface......Page 8
0. Introduction......Page 14
1. Linear Algebra and Projective Geometry......Page 20
1.1.1 Division rings......Page 21
1.1.2 Vector spaces over division rings......Page 23
1.1.3 Dual vector space......Page 27
1.2.1 Linear and partial linear spaces......Page 30
1.2.2 Projective spaces over division rings......Page 32
1.3.1 Definitions......Page 33
1.3.2 Mappings of Grassmannians induced by semilinear mappings......Page 34
1.3.3 Contragradient......Page 38
1.4.1 Main theorem and corollaries......Page 39
1.4.2 Proof of Theorem1.4......Page 41
1.4.3 Fundamental Theorem for normed spaces......Page 45
1.4.4 Proof of Theorem1.5......Page 46
1.5.1 Sesquilinear forms......Page 50
1.5.2 Reflexive forms......Page 51
1.5.3 Polarities......Page 53
2.1.1 Definition and examples......Page 56
2.1.2 Chamber complexes......Page 59
2.1.3 Grassmannians and Grassmann spaces......Page 60
2.2.1 Coxeter systems......Page 62
2.2.2 Coxeter complexes......Page 65
2.2.3 Three examples......Page 66
2.3.1 Definition and elementary properties......Page 68
2.3.2 Buildings and Tits systems......Page 70
2.3.3 Classical examples......Page 72
2.3.4 Spherical buildings......Page 75
2.3.5 Mappings of the chamber sets......Page 76
2.4 Mappings of Grassmannians......Page 78
2.5 Appendix: Gamma spaces......Page 80
3. Classical Grassmannians......Page 82
3.1 Elementary properties of Grassmann spaces......Page 83
3.2.1 Chow’s theorem......Page 88
3.2.2 Chow’s theoremfor linear spaces......Page 90
3.2.3 Applications of Chow’s theorem......Page 91
3.2.4 Opposite relation......Page 93
3.3.1 Basic properties......Page 96
3.3.2 Proof of Theorem3.8......Page 98
3.4.1 Results......Page 100
3.4.2 Proof of Theorem 3.10: First step......Page 102
3.4.3 Proof of Theorem 3.10: Second step......Page 106
3.5.1 Exchange spaces......Page 108
3.5.2 Grassmannians......Page 109
3.6 Matrix geometry and spine spaces......Page 113
3.7.1 Involutions and transvections......Page 115
3.7.2 Adjacency relation......Page 117
3.7.3 Chow’s theorem for linear involutions......Page 121
3.7.4 Proof of Theorem 3.15......Page 123
3.7.5 Automorphisms of the group GL(V )......Page 126
3.8.1 Adjacency relation......Page 127
3.8.2 Proof of Theorem 3.17......Page 129
3.8.4 Proof of Theorem 3.18......Page 131
4. Polar and Half-Spin Grassmannians......Page 136
4.1.1 Axioms and elementary properties......Page 138
4.1.2 Proof of Theorem 4.1......Page 139
4.1.3 Corollaries of Theorem 4.1......Page 142
4.1.4 Polar frames......Page 143
4.2.1 Polar Grassmannians......Page 146
4.2.2 Two types of polar spaces......Page 149
4.2.3 Half-spin Grassmannians......Page 151
4.3.1 Polar spaces associated with sesquilinear forms......Page 154
4.3.2 Polar spaces associated with quadratic forms......Page 159
4.3.3 Polar spaces of type D3......Page 160
4.3.4 Embeddings in projective spaces and classification......Page 162
4.4.2 Buildings of type Dn......Page 163
4.5.1 Polar Grassmann spaces......Page 164
4.5.2 Half-spin Grassmann spaces......Page 167
4.6.1 Chow’s theorem and its generalizations......Page 172
4.6.2 Weak adjacency on polar Grassmannians......Page 174
4.6.3 Proof of Theorem 4.8 for k < n-2......Page 175
4.6.4 Proof of Theorems 4.7 and 4.8......Page 177
4.6.5 Proof of Theorem 4.9......Page 178
4.6.6 Remarks......Page 185
4.7.1 Opposite relation on polar Grassmannians......Page 187
4.7.2 Opposite relation on half-spin Grassmannians......Page 188
4.8.1 Apartments in polar Grassmannians......Page 191
4.8.2 Apartments in half-spin Grassmannians......Page 194
4.8.3 Proof of Theorem 4.15......Page 197
4.9.1 Apartments preserving bijections......Page 199
4.9.2 Inexact subsets of polar Grassmannians......Page 200
4.9.3 Complement subsets of polar Grassmannians......Page 207
4.9.4 Inexact subsets of half-spin Grassmannians......Page 212
4.9.5 Proof of Theorem 4.16......Page 214
4.9.6 Embeddings......Page 215
4.9.7 Proof of Theorems 4.17 and 4.18......Page 216
Bibliography......Page 220
Index......Page 224