An edition of Grassmannians of classical buildings
(2010)
Grassmannians of classical buildings
by Mark Pankov
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Grassmannians of Classical Buildings
2010, World Scientific Publishing Co Pte Ltd
in English
9814317578 9789814317573
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Grassmannians of Classical Buildings
2010, World Scientific Publishing Co Pte Ltd
in English
1283144867 9781283144865
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Book Details
Table of Contents
Machine generated contents note: -- 1.1.
Vector spaces -- -- 1.1.1.
Division rings -- -- 1.1.2.
Vector spaces over division rings -- -- 1.1.3.
Dual vector space -- -- 1.2.
Projective spaces -- -- 1.2.1.
Linear and partial linear spaces -- -- 1.2.2.
Projective spaces over division rings -- -- 1.3.
Semilinear mappings -- -- 1.3.1.
Definitions -- -- 1.3.2.
Mappings of Grassmannians induced by semilinear mappings -- -- 1.3.3.
Contragradient -- -- 1.4.
Fundamental Theorem of Projective Geometry -- -- 1.4.1.
Main theorem and corollaries -- -- 1.4.2.
Proof of Theorem 1.4 -- -- 1.4.3.
Fundamental Theorem for normed spaces -- -- 1.4.4.
Proof of Theorem 1.5 -- -- 1.5.
Reflexive forms and polarities -- -- 1.5.1.
Sesquilinear forms -- -- 1.5.2.
Reflexive forms -- -- 1.5.3.
Polarities -- -- 2.1.
Simplicial complexes -- -- 2.1.1.
Definition and examples -- -- 2.1.2.
Chamber complexes -- -- 2.1.3.
Grassmannians and Grassmann spaces -- -- 2.2.
Coxeter systems and Coxeter complexes
2.2.1.
Coxeter systems -- -- 2.2.2.
Coxeter complexes -- -- 2.2.3.
Three examples -- -- 2.3.
Buildings -- -- 2.3.1.
Definition and elementary properties -- -- 2.3.2.
Buildings and Tits systems -- -- 2.3.3.
Classical examples -- -- 2.3.4.
Spherical buildings -- -- 2.3.5.
Mappings of the chamber sets -- -- 2.4.
Mappings of Grassmannians -- -- 2.5.
Appendix: Gamma spaces -- -- 3.1.
Elementary properties of Grassmann spaces -- -- 3.2.
Collineations of Grassmann spaces -- -- 3.2.1.
Chow's theorem -- -- 3.2.2.
Chow's theorem for linear spaces -- -- 3.2.3.
Applications of Chow's theorem -- -- 3.2.4.
Opposite relation -- -- 3.3.
Apartments -- -- 3.3.1.
Basic properties -- -- 3.3.2.
Proof of Theorem 3.8 -- -- 3.4.
Apartments preserving mappings -- -- 3.4.1.
Results -- -- 3.4.2.
Proof of Theorem 3.10: First step -- -- 3.4.3.
Proof of Theorem 3.10: Second step -- -- 3.5.
Grassmannians of exchange spaces -- -- 3.5.1.
Exchange spaces -- -- 3.5.2.
Grassmannians -- -- 3.6.
Matrix geometry and spine spaces -- -- 3.7.
Geometry of linear involutions
3.7.1.
Involutions and transvections -- -- 3.7.2.
Adjacency relation -- -- 3.7.3.
Chow's theorem for linear involutions -- -- 3.7.4.
Proof of Theorem 3.15 -- -- 3.7.5.
Automorphisms of the group GL(V) -- -- 3.8.
Grassmannians of infinite-dimensional vector spaces -- -- 3.8.1.
Adjacency relation -- -- 3.8.2.
Proof of Theorem 3.17 -- -- 3.8.3.
Base subsets -- -- 3.8.4.
Proof of Theorem 3.18 -- -- 4.1.
Polar spaces -- -- 4.1.1.
Axioms and elementary properties -- -- 4.1.2.
Proof of Theorem 4.1 -- -- 4.1.3.
Corollaries of Theorem 4.1 -- -- 4.1.4.
Polar frames -- -- 4.2.
Grassmannians -- -- 4.2.1.
Polar Grassmannians -- -- 4.2.2.
Two types of polar spaces -- -- 4.2.3.
Half-spin Grassmannians -- -- 4.3.
Examples -- -- 4.3.1.
Polar spaces associated with sesquilinear forms -- -- 4.3.2.
Polar spaces associated with quadratic forms -- -- 4.3.3.
Polar spaces of type D3 -- -- 4.3.4.
Embeddings in projective spaces and classification -- -- 4.4.
Polar buildings -- -- 4.4.1.
Buildings of type Cn -- -- 4.4.2.
Buildings of type Dn -- -- 4.5.
Elementary properties of Grassmann spaces
4.5.1.
Polar Grassmann spaces -- -- 4.5.2.
Half-spin Grassmann spaces -- -- 4.6.
Collineations -- -- 4.6.1.
Chow's theorem and its generalizations -- -- 4.6.2.
Weak adjacency on polar Grassmannians -- -- 4.6.3.
Proof of Theorem 4.8 for k < n [ -- ] 2 -- -- 4.6.4.
Proof of Theorems 4.7 and -- -- 4.6.5.
Proof of Theorem 4.9 -- -- 4.6.6.
Remarks -- -- 4.7.
Opposite relation -- -- 4.7.1.
Opposite relation on polar Grassmannians -- -- 4.7.2.
Opposite relation on half-spin Grassmannians -- -- 4.8.
Apartments -- -- 4.8.1.
Apartments in polar Grassmannians -- -- 4.8.2.
Apartments in half-spin Grassmannians -- -- 4.8.3.
Proof of Theorem 4.15 -- -- 4.9.
Apartments preserving mappings -- -- 4.9.1.
Apartments preserving bijections -- -- 4.9.2.
Inexact subsets of polar Grassmannians -- -- 4.9.3.
Complement subsets of polar Grassmannians -- -- 4.9.4.
Inexact subsets of half-spin Grassmannians -- -- 4.9.5.
Proof of Theorem 4.16 -- -- 4.9.6.
Embeddings -- -- 4.9.7.
Proof of Theorems 4.17 and 4.18.
Edition Notes
Includes bibliographical references (p. 207-210) and index.
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| August 2, 2020 | Edited by ImportBot | import existing book |
| July 25, 2012 | Created by LC Bot | import new book |