Grassmannians of classical buildings:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
World Scientific
2010
|
| Schriftenreihe: | Algebra and discrete mathematics
2 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 207 - 210 |
| Beschreibung: | XII, 212 S. graph. Darst. |
| ISBN: | 981431756X 9789814317566 |
Internformat
MARC
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| 300 | |a XII, 212 S. |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1804145823048531968 |
|---|---|
| adam_text | Titel: Grassmannians of classical buildings
Autor: Pankov, Mark
Jahr: 2010
Contents
Preface vii
0. Introduction 1
1. Linear Algebra and Projective Geometry 7
1.1 Vector spaces......................... 8
1.1.1 Division rings.................... 8
1.1.2 Vector spaces over division rings.......... 10
1.1.3 Dual vector space.................. 14
1.2 Projective spaces....................... 17
1.2.1 Linear and partial linear spaces.......... 17
1.2.2 Projective spaces over division rings........ 19
1.3 Semilinear mappings..................... 20
1.3.1 Definitions...................... 20
1.3.2 Mappings of Grassmannians induced by semilinear
mappings....................... 21
1.3.3 Contragradient.................... 25
1.4 Fundamental Theorem of Projective Geometry...... 26
1.4.1 Main theorem and corollaries............ 26
1.4.2 Proof of Theorem 1.4................ 28
1.4.3 Fundamental Theorem for normed spaces..... 32
1.4.4 Proof of Theorem 1.5................ 33
1.5 Reflexive forms and polarities................ 37
1.5.1 Sesquilinear forms.................. 37
1.5.2 Reflexive forms ................... 38
1.5.3 Polarities....................... 40
x Gmssmannians of Classical Buildings
2. Buildings and Grassmannians 43
2.1 Simplicial complexes..................... 43
2.1.1 Definition and examples .............. 43
2.1.2 Chamber complexes................. 46
2.1.3 Grassmannians and Grassmann spaces...... 47
2.2 Coxeter systems and Coxeter complexes.......... 49
2.2.1 Coxeter systems................... 49
2.2.2 Coxeter complexes.................. 52
2.2.3 Three examples................... 53
2.3 Buildings........................... 55
2.3.1 Definition and elementary properties....... 55
2.3.2 Buildings and Tits systems............. 57
2.3.3 Classical examples.................. 59
2.3.4 Spherical buildings ................. 62
2.3.5 Mappings of the chamber sets........... 63
2.4 Mappings of Grassmannians................. 65
2.5 Appendix: Gamma spaces.................. 67
3. Classical Grassmannians 69
3.1 Elementary properties of Grassmann spaces........ 70
3.2 Collineations of Grassmann spaces............. 75
3.2.1 Chow s theorem................... 75
3.2.2 Chow s theorem for linear spaces.......... 77
3.2.3 Applications of Chow s theorem.......... 78
3.2.4 Opposite relation.................. 80
3.3 Apartments.......................... 83
3.3.1 Basic properties................... 83
3.3.2 Proof of Theorem 3.8................ 85
3.4 Apartments preserving mappings.............. 87
3.4.1 Results........................ 87
3.4.2 Proof of Theorem 3.10: First step......... 89
3.4.3 Proof of Theorem 3.10: Second step........ 93
3.5 Grassmannians of exchange spaces............. 95
3.5.1 Exchange spaces................... 95
3.5.2 Grassmannians.................... 96
3.6 Matrix geometry and spine spaces............. 100
3.7 Geometry of linear involutions ............... 102
3.7.1 Involutions and transvections............ 102
Contents xi
3.7.2 Adjacency relation.................. 104
3.7.3 Chow s theorem for linear involutions....... 108
3.7.4 Proof of Theorem 3.15 ............... 110
3.7.5 Automorphisms of the group GL(F)........ 113
3.8 Grassmannians of infinite-dimensional vector spaces ... 114
3.8.1 Adjacency relation.................. 114
3.8.2 Proof of Theorem 3.17............... 116
3.8.3 Base subsets..................... 118
3.8.4 Proof of Theorem 3.18 ............... 118
4. Polar and Half-Spin Grassmannians 123
4.1 Polar spaces.......................... 125
4.1.1 Axioms and elementary properties......... 125
4.1.2 Proof of Theorem 4.1................ 126
4.1.3 Corollaries of Theorem 4.1............. 129
4.1.4 Polar frames..................... 130
4.2 Grassmannians........................ 133
4.2.1 Polar Grassmannians................ 133
4.2.2 Two types of polar spaces ............. 136
4.2.3 Half-spin Grassmannians.............. 138
4.3 Examples........................... 141
4.3.1 Polar spaces associated with sesquilinear forms . . 141
4.3.2 Polar spaces associated with quadratic forms . . . 146
4.3.3 Polar spaces of type D3............... 147
4.3.4 Embeddings in projective spaces and classification 149
4.4 Polar buildings........................ 150
4.4.1 Buildings of type Cn................. 150
4.4.2 Buildings of type Dn ................ 150
4.5 Elementary properties of Grassmann spaces........ 151
4.5.1 Polar Grassmann spaces .............. 151
4.5.2 Half-spin Grassmann spaces............ 154
4.6 Collineations......................... 159
4.6.1 Chow s theorem and its generalizations...... 159
4.6.2 Weak adjacency on polar Grassmannians..... 161
4.6.3 Proof of Theorem 4.8 for k ç - 2........ 162
4.6.4 Proof of Theorems 4.7 and 4.8........... 164
4.6.5 Proof of Theorem 4.9................ 165
4.6.6 Remarks....................... 172
4.7 Opposite relation....................... 174
xii Grassmannians of Classical Buildings
4.7.1 Opposite relation on polar Grassmannians .... 174
4.7.2 Opposite relation on half-spin Grassmannians . . 175
4.8 Apartments.......................... 178
4.8.1 Apartments in polar Grassmannians........ 178
4.8.2 Apartments in half-spin Grassmannians...... 181
4.8.3 Proof of Theorem 4.15 ............... 184
4.9 Apartments preserving mappings.............. 186
4.9.1 Apartments preserving bijections ......... 186
4.9.2 Inexact subsets of polar Grassmannians...... 187
4.9.3 Complement subsets of polar Grassmannians . . . 194
4.9.4 Inexact subsets of half-spin Grassmannians .... 199
4.9.5 Proof of Theorem 4.16 ............... 201
4.9.6 Embeddings..................... 202
4.9.7 Proof of Theorems 4.17 and 4.18.......... 203
Bibliography 207
Index 211
|
| any_adam_object | 1 |
| author | Pankov, Mark ca. 20./21. Jh |
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| id | DE-604.BV039109925 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T23:25:34Z |
| institution | BVB |
| isbn | 981431756X 9789814317566 |
| language | English |
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| publisher | World Scientific |
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| series | Algebra and discrete mathematics |
| series2 | Algebra and discrete mathematics |
| spelling | Pankov, Mark ca. 20./21. Jh. Verfasser (DE-588)1033189499 aut Grassmannians of classical buildings Mark Pankov New Jersey [u.a.] World Scientific 2010 XII, 212 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algebra and discrete mathematics 2 Literaturverz. S. 207 - 210 Gebäude Mathematik (DE-588)4123258-6 gnd rswk-swf Graßmann-Mannigfaltigkeit (DE-588)4158085-0 gnd rswk-swf Gruppe Mathematik (DE-588)4022379-6 gnd rswk-swf Graßmann-Mannigfaltigkeit (DE-588)4158085-0 s Gruppe Mathematik (DE-588)4022379-6 s Gebäude Mathematik (DE-588)4123258-6 s DE-604 Algebra and discrete mathematics 2 (DE-604)BV039152082 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022653663&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pankov, Mark ca. 20./21. Jh Grassmannians of classical buildings Algebra and discrete mathematics Gebäude Mathematik (DE-588)4123258-6 gnd Graßmann-Mannigfaltigkeit (DE-588)4158085-0 gnd Gruppe Mathematik (DE-588)4022379-6 gnd |
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| title | Grassmannians of classical buildings |
| title_auth | Grassmannians of classical buildings |
| title_exact_search | Grassmannians of classical buildings |
| title_full | Grassmannians of classical buildings Mark Pankov |
| title_fullStr | Grassmannians of classical buildings Mark Pankov |
| title_full_unstemmed | Grassmannians of classical buildings Mark Pankov |
| title_short | Grassmannians of classical buildings |
| title_sort | grassmannians of classical buildings |
| topic | Gebäude Mathematik (DE-588)4123258-6 gnd Graßmann-Mannigfaltigkeit (DE-588)4158085-0 gnd Gruppe Mathematik (DE-588)4022379-6 gnd |
| topic_facet | Gebäude Mathematik Graßmann-Mannigfaltigkeit Gruppe Mathematik |
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