Grassmannians of classical buildings
By: Mark Pankov
Series: Algebra and discrete mathematics -- v. 2; Algebra and discrete mathematics (World Scientific (Firm)) -- v. 2.
Publisher: World Scientific
Published: 2010
Place of publication: Singapore; Hackensack, NJ
Language: eng
Number of pages: 212
ISBN-13: 9789814317566
Subjects: Mathematics; Grassmann manifolds; Architecture
Notes: Includes bibliographical references (p. 207-210) and index.
Table of contents: title: Machine generated contents note: -- 1.1.; title: Vector spaces -- -- 1.1.1.; title: Division rings -- -- 1.1.2.; title: Vector spaces over division rings -- -- 1.1.3.; title: Dual vector space -- -- 1.2.; title: Projective spaces -- -- 1.2.1.; title: Linear and partial linear spaces -- -- 1.2.2.; title: Projective spaces over division rings -- -- 1.3.; title: Semilinear mappings -- -- 1.3.1.; title: Definitions -- -- 1.3.2.; title: Mappings of Grassmannians induced by semilinear mappings -- -- 1.3.3.; title: Contragradient -- -- 1.4.; title: Fundamental Theorem of Projective Geometry -- -- 1.4.1.; title: Main theorem and corollaries -- -- 1.4.2.; title: Proof of Theorem 1.4 -- -- 1.4.3.; title: Fundamental Theorem for normed spaces -- -- 1.4.4.; title: Proof of Theorem 1.5 -- -- 1.5.; title: Reflexive forms and polarities -- -- 1.5.1.; title: Sesquilinear forms -- -- 1.5.2.; title: Reflexive forms -- -- 1.5.3.; title: Polarities -- -- 2.1.; title: Simplicial complexes -- -- 2.1.1.; title: Definition and examples -- -- 2.1.2.; title: Chamber complexes -- -- 2.1.3.; title: Grassmannians and Grassmann spaces -- -- 2.2.; title: Coxeter systems and Coxeter complexes; title: 2.2.1.; title: Coxeter systems -- -- 2.2.2.; title: Coxeter complexes -- -- 2.2.3.; title: Three examples -- -- 2.3.; title: Buildings -- -- 2.3.1.; title: Definition and elementary properties -- -- 2.3.2.; title: Buildings and Tits systems -- -- 2.3.3.; title: Classical examples -- -- 2.3.4.; title: Spherical buildings -- -- 2.3.5.; title: Mappings of the chamber sets -- -- 2.4.; title: Mappings of Grassmannians -- -- 2.5.; title: Appendix: Gamma spaces -- -- 3.1.; title: Elementary properties of Grassmann spaces -- -- 3.2.; title: Collineations of Grassmann spaces -- -- 3.2.1.; title: Chow's theorem -- -- 3.2.2.; title: Chow's theorem for linear spaces -- -- 3.2.3.; title: Applications of Chow's theorem -- -- 3.2.4.; title: Opposite relation -- -- 3.3.; title: Apartments -- -- 3.3.1.; title: Basic properties -- -- 3.3.2.; title: Proof of Theorem 3.8 -- -- 3.4.; title: Apartments preserving mappings -- -- 3.4.1.; title: Results -- -- 3.4.2.; title: Proof of Theorem 3.10: First step -- -- 3.4.3.; title: Proof of Theorem 3.10: Second step -- -- 3.5.; title: Grassmannians of exchange spaces -- -- 3.5.1.; title: Exchange spaces -- -- 3.5.2.; title: Grassmannians -- -- 3.6.; title: Matrix geometry and spine spaces -- -- 3.7.; title: Geometry of linear involutions; title: 3.7.1.; title: Involutions and transvections -- -- 3.7.2.; title: Adjacency relation -- -- 3.7.3.; title: Chow's theorem for linear involutions -- -- 3.7.4.; title: Proof of Theorem 3.15 -- -- 3.7.5.; title: Automorphisms of the group GL(V) -- -- 3.8.; title: Grassmannians of infinite-dimensional vector spaces -- -- 3.8.1.; title: Adjacency relation -- -- 3.8.2.; title: Proof of Theorem 3.17 -- -- 3.8.3.; title: Base subsets -- -- 3.8.4.; title: Proof of Theorem 3.18 -- -- 4.1.; title: Polar spaces -- -- 4.1.1.; title: Axioms and elementary properties -- -- 4.1.2.; title: Proof of Theorem 4.1 -- -- 4.1.3.; title: Corollaries of Theorem 4.1 -- -- 4.1.4.; title: Polar frames -- -- 4.2.; title: Grassmannians -- -- 4.2.1.; title: Polar Grassmannians -- -- 4.2.2.; title: Two types of polar spaces -- -- 4.2.3.; title: Half-spin Grassmannians -- -- 4.3.; title: Examples -- -- 4.3.1.; title: Polar spaces associated with sesquilinear forms -- -- 4.3.2.; title: Polar spaces associated with quadratic forms -- -- 4.3.3.; title: Polar spaces of type D3 -- -- 4.3.4.; title: Embeddings in projective spaces and classification -- -- 4.4.; title: Polar buildings -- -- 4.4.1.; title: Buildings of type Cn -- -- 4.4.2.; title: Buildings of type Dn -- -- 4.5.; title: Elementary properties of Grassmann spaces; title: 4.5.1.; title: Polar Grassmann spaces -- -- 4.5.2.; title: Half-spin Grassmann spaces -- -- 4.6.; title: Collineations -- -- 4.6.1.; title: Chow's theorem and its generalizations -- -- 4.6.2.; title: Weak adjacency on polar Grassmannians -- -- 4.6.3.; title: Proof of Theorem 4.8 for k < n [ -- ] 2 -- -- 4.6.4.; title: Proof of Theorems 4.7 and -- -- 4.6.5.; title: Proof of Theorem 4.9 -- -- 4.6.6.; title: Remarks -- -- 4.7.; title: Opposite relation -- -- 4.7.1.; title: Opposite relation on polar Grassmannians -- -- 4.7.2.; title: Opposite relation on half-spin Grassmannians -- -- 4.8.; title: Apartments -- -- 4.8.1.; title: Apartments in polar Grassmannians -- -- 4.8.2.; title: Apartments in half-spin Grassmannians -- -- 4.8.3.; title: Proof of Theorem 4.15 -- -- 4.9.; title: Apartments preserving mappings -- -- 4.9.1.; title: Apartments preserving bijections -- -- 4.9.2.; title: Inexact subsets of polar Grassmannians -- -- 4.9.3.; title: Complement subsets of polar Grassmannians -- -- 4.9.4.; title: Inexact subsets of half-spin Grassmannians -- -- 4.9.5.; title: Proof of Theorem 4.16 -- -- 4.9.6.; title: Embeddings -- -- 4.9.7.; title: Proof of Theorems 4.17 and 4.18.