Reflection Groups And Coxeter Groups
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Reflection Groups And Coxeter Groups
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Author : James E. Humphreys
language : en
Publisher: Cambridge University Press
Release Date : 1992-10
Reflection Groups And Coxeter Groups written by James E. Humphreys and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-10 with Mathematics categories.
This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.
Finite Reflection Groups
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Author : L.C. Grove
language : en
Publisher: Springer Science & Business Media
Release Date : 1996-06-20
Finite Reflection Groups written by L.C. Grove and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-06-20 with Mathematics categories.
Chapter 1 introduces some of the terminology and notation used later and indicates prerequisites. Chapter 2 gives a reasonably thorough account of all finite subgroups of the orthogonal groups in two and three dimensions. The presentation is somewhat less formal than in succeeding chapters. For instance, the existence of the icosahedron is accepted as an empirical fact, and no formal proof of existence is included. Throughout most of Chapter 2 we do not distinguish between groups that are "geo metrically indistinguishable," that is, conjugate in the orthogonal group. Very little of the material in Chapter 2 is actually required for the sub sequent chapters, but it serves two important purposes: It aids in the development of geometrical insight, and it serves as a source of illustrative examples. There is a discussion offundamental regions in Chapter 3. Chapter 4 provides a correspondence between fundamental reflections and funda mental regions via a discussion of root systems. The actual classification and construction of finite reflection groups takes place in Chapter 5. where we have in part followed the methods of E. Witt and B. L. van der Waerden. Generators and relations for finite reflection groups are discussed in Chapter 6. There are historical remarks and suggestions for further reading in a Post lude.
Reflection Groups And Invariant Theory
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Author : Richard Kane
language : en
Publisher: Springer Science & Business Media
Release Date : 2001-06-21
Reflection Groups And Invariant Theory written by Richard Kane and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-06-21 with Mathematics categories.
Reflection groups and invariant theory is a branch of mathematics that lies at the intersection between geometry and algebra. The book contains a deep and elegant theory, evolved from various graduate courses given by the author over the past 10 years.
Combinatorics Of Coxeter Groups
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Author : Anders Bjorner
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-02-25
Combinatorics Of Coxeter Groups written by Anders Bjorner and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-25 with Mathematics categories.
Includes a rich variety of exercises to accompany the exposition of Coxeter groups Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups
The Geometry And Topology Of Coxeter Groups
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Author : Michael Davis
language : en
Publisher: Princeton University Press
Release Date : 2008
The Geometry And Topology Of Coxeter Groups written by Michael Davis and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.
Unitary Reflection Groups
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Author : Gustav I. Lehrer
language : en
Publisher: Cambridge University Press
Release Date : 2009-08-13
Unitary Reflection Groups written by Gustav I. Lehrer and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-08-13 with Mathematics categories.
A unitary reflection is a linear transformation of a complex vector space that fixes each point in a hyperplane. Intuitively, it resembles the transformation an image undergoes when it is viewed through a kaleidoscope, or an arrangement of mirrors. This book gives a complete classification of all finite groups which are generated by unitary reflections, using the method of line systems. Irreducible groups are studied in detail, and are identified with finite linear groups. The new invariant theoretic proof of Steinberg's fixed point theorem is treated fully. The same approach is used to develop the theory of eigenspaces of elements of reflection groups and their twisted analogues. This includes an extension of Springer's theory of regular elements to reflection cosets. An appendix outlines links to representation theory, topology and mathematical physics. Containing over 100 exercises, ranging in difficulty from elementary to research level, this book is ideal for honours and graduate students, or for researchers in algebra, topology and mathematical physics. Book jacket.
Mirrors And Reflections
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Author : Alexandre V. Borovik
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-11-07
Mirrors And Reflections written by Alexandre V. Borovik and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-11-07 with Mathematics categories.
This graduate/advanced undergraduate textbook contains a systematic and elementary treatment of finite groups generated by reflections. The approach is based on fundamental geometric considerations in Coxeter complexes, and emphasizes the intuitive geometric aspects of the theory of reflection groups. Key features include: many important concepts in the proofs are illustrated in simple drawings, which give easy access to the theory; a large number of exercises at various levels of difficulty; some Euclidean geometry is included along with the theory of convex polyhedra; no prerequisites are necessary beyond the basic concepts of linear algebra and group theory; and a good index and bibliography The exposition is directed at advanced undergraduates and first-year graduate students.
Coxeter Groups And Hopf Algebras
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Author : Marcelo Aguiar
language : en
Publisher: American Mathematical Soc.
Release Date : 2006
Coxeter Groups And Hopf Algebras written by Marcelo Aguiar and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Education categories.
An important idea in the work of G.-C. Rota is that certain combinatorial objects give rise to Hopf algebras that reflect the manner in which these objects compose and decompose. Recent work has seen the emergence of several interesting Hopf algebras of this kind, which connect diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This monograph presents a novel geometric approach using Coxeter complexes and the projection maps of Tits for constructing and studying many of these objects as well as new ones. The first three chapters introduce the necessary background ideas making this work accessible to advanced graduate students. The later chapters culminate in a unified and conceptual construction of several Hopf algebras based on combinatorial objects which emerge naturally from the geometric viewpoint. This work lays a foundation and provides new insights for further development of the subject.
Coxeter Matroids
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Author : Alexandre V. Borovik
language : en
Publisher: Birkhäuser
Release Date : 2003-07-11
Coxeter Matroids written by Alexandre V. Borovik and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-07-11 with Mathematics categories.
Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry, and "Coxeter Matroids" provides an intuitive and interdisciplinary treatment of their theory. In this text, matroids are examined in terms of symmetric and finite reflection groups; also, symplectic matroids and the more general coxeter matroids are carefully developed. The Gelfand-Serganova theorem, which allows for the geometric interpretation of matroids as convex polytopes with certain symmetry properties, is presented, and in the final chapter, matroid representations and combinatorial flag varieties are discussed. With its excellent bibliography and index and ample references to current research, this work will be useful for graduate students and research mathematicians.
Geometry Of Coxeter Groups
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Author : Howard Hiller
language : en
Publisher: Pitman Publishing
Release Date : 1982
Geometry Of Coxeter Groups written by Howard Hiller and has been published by Pitman Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 1982 with Mathematics categories.