Overview

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 formulated in terms of graph theory, that is, Grassmann graphs, graph embeddings, and isometric embeddings. In addition, some relations to linear codes will be described. Graduate students and researchers will find this volume to be self-contained with many examples.Contents: Semilinear Mappings: Division Rings and Their HomomorphismsVector Spaces Over Division RingsSemilinear MappingsSemilinear EmbeddingsMappings of Grassmannians Induced by Semilinear EmbeddingsKreuzer's ExampleDualityCharacterization of Strong Semilinear Embeddings Projective Geometry and Linear Codes: Projective SpacesFundamental Theorem of Projective GeometryProof of Theorem 1.2 m -independent Subsets in Projective SpacesPGL-subsetsGeneralized MacWilliams TheoremLinear Codes Isometric Embeddings of Grassmann Graphs: Graph TheoryElementary Properties of Grassmann GraphsEmbeddingsIsometric EmbeddingsProof of Theorem 3.1Equivalence of Isometric EmbeddingsLinearly Rigid Isometric EmbeddingsRemarks on Non-isometric EmbeddingsSome Results Related to Chow's TheoremHuang's Theorem Johnson Graph in Grassmann Graph: Johnson GraphIsometric Embeddings of Johnson Graphs in Grassmann GraphsProof of Theorem 4.2Classification Problem and Relations to Linear CodesCharacterizations of Apartments in Building Grassmannians Characterization of Isometric Embeddings: Main Result, Corollaries and RemarksCharacterization of DistanceConnectedness of the Apartment GraphIntersections of J ( n, k )-subsets of Different TypesProof of Theorem 5.1 Semilinear Mappings of Exterior Powers: Exterior PowersGrassmanniansGrassmann CodesReadership: Graduate students and researchers interested in the field of semilinear embeddings.

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