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.
The first part of this book introduces the Schubert Cells and varieties of the general linear group Gl (k^(r+1)) over a field k according to Ehresmann geometric way. Smooth resolutions for these varieties are constructed in terms of Flag Configurations in k^(r+1) given by linear graphs called Minimal Galleries. In the second part, Schubert Schemes, the Universal Schubert Scheme and their Canonical Smooth Resolution, in terms of the incidence relation in a Tits relative building are constructed for a Reductive Group Scheme as in Grothendieck's SGAIII. This is a topic where algebra and algebraic geometry, combinatorics, and group theory interact in unusual and deep ways.
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.
The book deals with fundamental structural aspects of algebraic and simple groups, Coxeter groups and the related geometries and buildings. All contributing authors are very active researchers in the topics related to the theme of the book. Some of the articles provide the latest developments in the subject; some provide an overview of the current status of some important problems in this area; some survey an area highlighting the current developments; and some provide an exposition of an area to collect problems and conjectures. It is hoped that these articles would be helpful to a beginner to start independent research on any of these topics, as well as to an expert to know some of the latest developments or to consider some problems for investigation.
Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. This book gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group.
It is eleven years since the First Edition of Geometry of Crystallographic Groups appeared. This Second Edition expands on the first, providing details of a new result of automorphism of crystallographic groups, and on Hantzsche-Wendt groups/manifolds.Crystalographic groups are groups which act via isometries on some n-dimensional Euclidean space, so-named because in three dimensions they occur as the symmetry groups of a crystal. There are short introductions to the theme before every chapter, and a list of conjectures and open projects at the end of the book.Geometry of Crystallographic Groups is suitable as a textbook for students, containing basic theory of crystallographic groups. It is also suitable for researchers in the field, discussing in its second half more advanced and recent topics.
This textbook presents fundamental topics in discrete mathematics introduced from the perspectives of a pure mathematician and an applied computer scientist. The synergy between the two complementary perspectives is seen throughout the book; key concepts are motivated and explained through real-world examples, and yet are still formalized with mathematical rigor. The book is an excellent introduction to discrete mathematics for computer science, software engineering, and mathematics students.The first author is a leading mathematician in the area of logic, computability, and theoretical computer science, with more than 25 years of teaching and research experience. The second author is a computer science PhD student at the University of Washington specializing in database systems. The father-and-daughter team merges two different views to create a unified book for students interested in learning discrete mathematics, the connections between discrete mathematics and computer science, and the mathematical foundations of computer science.Readers will learn how to formally define abstract concepts, reason about objects (such as programs, graphs and numbers), investigate properties of algorithms, and prove their correctness. The textbook studies several well-known algorithmic problems including the path problem for graphs and finding the greatest common divisor, inductive definitions, proofs of correctness of algorithms via loop invariants and induction, the basics of formal methods such as propositional logic, finite state machines, counting, probability, as well as the foundations of databases such as relational calculus.
