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Geometry of semilinear embeddings : relations to graphs and codes /
"This volume covers semilinear embeddings of vector spaces over division rings and the associated mappings of Grassmannians. In contrast to classical books, we consider a more general class of semilinear mappings and show that this class is important. A large portion of the material will be for...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
New Jersey :
World Scientific,
[2015]
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; 1. Semilinear mappings; 1.1 Division rings and their homomorphisms; 1.2 Vector spaces over division rings; 1.3 Semilinear mappings; 1.4 Semilinear embeddings; 1.5 Mappings of Grassmannians induced by semilinear embeddings; 1.6 Kreuzer's example; 1.7 Duality; 1.8 Characterization of strong semilinear embeddings; 2. Projective Geometry and linear codes; 2.1 Projective spaces; 2.2 Fundamental Theorem of Projective Geometry; 2.3 Proof of Theorem 1.2; 2.4 m-independent subsets in projective spaces; 2.5 PGL-subsets; 2.6 Generalized MacWilliams theorem; 2.7 Linear codes.
- 3. Isometric embeddings of Grassmann graphs3.1 Graph theory; 3.2 Elementary properties of Grassmann graphs; 3.3 Embeddings; 3.4 Isometric embeddings; 3.5 Proof of Theorem 3.1; 3.6 Equivalence of isometric embeddings; 3.7 Linearly rigid isometric embeddings; 3.8 Remarks on non-isometric embeddings; 3.9 Some results related to Chow's theorem; 3.10 Huang's theorem; 3.10.1 Proof of Theorem 3.2 for n = 2k; 3.10.2 Proof of Theorem 3.2 for n 6= 2k; 4. Johnson graph in Grassmann graph; 4.1 Johnson graph; 4.2 Isometric embeddings of Johnson graphs in Grassmann graphs; 4.3 Proof of Theorem 4.2.
- 4.4 Classification problem and relations to linear codes4.5 Characterizations of apartments in building Grassmannians; 5. Characterization of isometric embeddings; 5.1 Main result, corollaries and remarks; 5.2 Characterization of distance; 5.3 Connectedness of the apartment graph; 5.4 Intersections of J(n, k)-subsets of different types; 5.5 Proof of Theorem 5.1; 6. Semilinear mappings of exterior powers; 6.1 Exterior powers; 6.2 Grassmannians; 6.3 Grassmann codes; Bibliography; Index.