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Grassmannians Of Classical Buildings : Algebra And Discrete Mathematics - Mark Pankov

Grassmannians Of Classical Buildings

Algebra And Discrete Mathematics

Hardcover Published: 30th January 2011
ISBN: 9789814317566
Number Of Pages: 224

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Buildings are combinatorial constructions successfully exploited to study groups of various types. The vertex set of a building can be naturally decomposed into subsets called Grassmannians. The book contains both classical and more recent results on Grassmannians of buildings of classical types. It gives a modern interpretation of some classical results from the geometry of linear groups. The presented methods are applied to some geometric constructions non-related to buildings - Grassmannians of infinite-dimensional vector spaces and the sets of conjugate linear involutions.The book is self-contained and the requirement for the reader is a knowledge of basic algebra and graph theory. This makes it very suitable for use in a course for graduate students.

Prefacep. vii
Introductionp. 1
Linear Algebra and Projective Geometryp. 7
Vector spacesp. 8
Division ringsp. 8
Vector spaces over division ringsp. 10
Dual vector spacep. 14
Projective spacesp. 17
Linear and partial linear spacesp. 17
Projective spaces over division ringsp. 19
Semilinear mappingsp. 20
Definitionsp. 20
Mappings of Grassmannians induced by semilinear mappingsp. 21
Contragradientp. 25
Fundamental Theorem of Projective Geometryp. 26
Main theorem and corollariesp. 26
Proof of Theorem 1.4p. 28
Fundamental Theorem for normed spacesp. 32
Proof of Theorem 1.5p. 33
Reflexive forms and polaritiesp. 37
Sesquilinear formsp. 37
Reflexive formsp. 38
Polaritiesp. 40
Buildings and Grassmanniansp. 43
Simplicial complexesp. 43
Definition and examplesp. 43
Chamber complexesp. 46
Grassmannians and Grassmann spacesp. 47
Coxeter systems and Coxeter complexesp. 49
Coxeter systemsp. 49
Coxeter complexesp. 52
Three examplesp. 53
Buildingsp. 55
Definition and elementary propertiesp. 55
Buildings and Tits systemsp. 57
Classical examplesp. 59
Spherical buildingsp. 62
Mappings of the chamber setsp. 63
Mappings of Grassmanniansp. 65
Appendix: Gamma spacesp. 67
Classical Grassmanniansp. 69
Elementary properties of Grassmann spacesp. 70
Collineations of Grassmann spacesp. 75
Chow's theoremp. 75
Chow's theorem for linear spacesp. 77
Applications of Chow's theoremp. 78
Opposite relationp. 80
Apartmentsp. 83
Basic propertiesp. 83
Proof of Theorem 3.8p. 85
Apartments preserving mappingsp. 87
Resultsp. 87
Proof of Theorem 3.10: First stepp. 89
Proof of Theorem 3.10: Second stepp. 93
Grassmannians of exchange spacesp. 95
Exchange spacesp. 95
Grassmanniansp. 96
Matrix geometry and spine spacesp. 100
Geometry of linear involutionsp. 102
Involutions and transvectionsp. 102
Adjacency relationp. 104
Chow's theorem for linear involutionsp. 108
Proof of Theorem 3.15p. 110
Automorphisms of the group GL(V)p. 113
Grassmannians of infinite-dimensional vector spacesp. 114
Adjacency relationp. 114
Proof of Theorem 3.17p. 116
Base subsetsp. 118
Proof of Theorem 3.18p. 118
Polar and Half-Spin Grassmanniansp. 123
Polar spacesp. 125
Axioms and elementary propertiesp. 125
Proof of Theorem 4.1p. 126
Corollaries of Theorem 4.1p. 129
Polar framesp. 130
Grassmanniansp. 133
Polar Grassmanniansp. 133
Two types of polar spacesp. 136
Half-spin Grassmanniansp. 138
Examplesp. 141
Polar spaces associated with sesquilinear formsp. 141
Polar spaces associated with quadratic formsp. 146
Polar spaces of type D3p. 147
Embeddings in projective spaces and classificationp. 149
Polar buildingsp. 150
Buildings of type Cnp. 150
Buildings of type Dnp. 150
Elementary properties of Grassmann spacesp. 151
Polar Grassmann spacesp. 151
Half-spin Grassmann spacesp. 154
Collineationsp. 159
Chow's theorem and its generalizationsp. 159
Weak adjacency on polar Grassmanniansp. 161
Proof of Theorem 4.8 for k < n - 2p. 162
Proof of Theorems 4.7 and 4.8p. 164
Proof of Theorem 4.9p. 165
Remarksp. 172
Opposite relationp. 174
Opposite relation on polar Grassmanniansp. 174
Opposite relation on half-spin Grassmanniansp. 175
Apartmentsp. 178
Apartments in polar Grassmanniansp. 178
Apartments in half-spin Grassmanniansp. 181
Proof of Theorem 4.15p. 184
Apartments preserving mappingsp. 186
Apartments preserving bijectionsp. 186
Inexact subsets of polar Grassmanniansp. 187
Complement subsets of polar Grassmanniansp. 194
Inexact subsets of half-spin Grassmanniansp. 199
Proof of Theorem 4.16p. 201
Embeddingsp. 202
Proof of Theorems 4.17 and 4.18p. 203
Bibliographyp. 207
Indexp. 211
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789814317566
ISBN-10: 981431756X
Series: Algebra And Discrete Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 224
Published: 30th January 2011
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 22.86 x 15.49  x 2.03
Weight (kg): 0.52