Reflection Groups And Coxeter Groups

Author : James E. Humphreys
Publisher : Cambridge University Press
Page : 222 pages
File Size : 42,9 Mb
Release : 1992-10
Category : Mathematics
ISBN : 0521436133

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Reflection Groups and Coxeter Groups by James E. Humphreys Pdf

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.

Author : L.C. Grove,C.T. Benson
Publisher : Springer Science & Business Media
Page : 142 pages
File Size : 41,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475718690

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Finite Reflection Groups by L.C. Grove,C.T. Benson Pdf

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.

Author : Michael Davis
Publisher : Princeton University Press
Page : 601 pages
File Size : 44,8 Mb
Release : 2008
Category : Mathematics
ISBN : 9780691131382

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The Geometry and Topology of Coxeter Groups by Michael Davis Pdf

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.

Author : Richard Kane
Publisher : Springer Science & Business Media
Page : 382 pages
File Size : 43,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475735420

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Reflection Groups and Invariant Theory by Richard Kane Pdf

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.

Author : Anders Bjorner,Francesco Brenti
Publisher : Springer Science & Business Media
Page : 371 pages
File Size : 44,5 Mb
Release : 2006-02-25
Category : Mathematics
ISBN : 9783540275961

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Combinatorics of Coxeter Groups by Anders Bjorner,Francesco Brenti Pdf

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

Author : Alexandre V. Borovik,Anna Borovik
Publisher : Springer Science & Business Media
Page : 172 pages
File Size : 52,7 Mb
Release : 2009-11-07
Category : Mathematics
ISBN : 9780387790664

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Mirrors and Reflections by Alexandre V. Borovik,Anna Borovik Pdf

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.

Author : Alexandre V. Borovik,Israel M. Gelfand,Neil White
Publisher : Springer Science & Business Media
Page : 282 pages
File Size : 46,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461220664

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Coxeter Matroids by Alexandre V. Borovik,Israel M. Gelfand,Neil White Pdf

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.

Author : Harold Scott Macdonald Coxeter,William O. J. Moser
Publisher : Springer Science & Business Media
Page : 163 pages
File Size : 51,9 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783662257395

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Generators and Relations for Discrete Groups by Harold Scott Macdonald Coxeter,William O. J. Moser Pdf

When we began to consider the scope of this book, we envisaged a catalogue supplying at least one abstract definition for any finitely generated group that the reader might propose. But we soon realized that more or less arbitrary restrictions are necessary, because interesting groups are so numerous. For permutation groups of degree 8 or less (i. e., subgroups of e ), the reader cannot do better than consult the 8 tables of JosEPHINE BuRNS (1915), while keeping an eye open for misprints. Our own tables (on pages 134-143) deal with groups of low order, finiteandinfinite groups of congruent transformations, symmetric and alternating groups, linear fractional groups, and groups generated by reflections in real Euclidean space of any number of dimensions. The best substitute foramoreextensive catalogue is the description (in Chapter 2) of a method whereby the reader can easily work out his own abstract definition for almost any given finite group. This method is sufficiently mechanical for the use of an electronic computer. There is also a topological method (Chapter 3), suitable not only for groups of low order but also for some infinite groups. This involves choosing a set of generators, constructing a certain graph (the Cayley diagram or DEHNsehe Gruppenbild), and embedding the graph into a surface. Cases in which the surface is a sphere or a plane are described in Chapter 4, where we obtain algebraically, and verify topologically, an abstract definition for each of the 17 space groups of two-dimensional crystallography.

Author : Gustav I. Lehrer,Donald E. Taylor
Publisher : Cambridge University Press
Page : 303 pages
File Size : 43,6 Mb
Release : 2009-08-13
Category : Mathematics
ISBN : 9780521749893

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Unitary Reflection Groups by Gustav I. Lehrer,Donald E. Taylor Pdf

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.

Author : Ben Elias,Shotaro Makisumi,Ulrich Thiel,Geordie Williamson
Publisher : Springer Nature
Page : 588 pages
File Size : 54,5 Mb
Release : 2020-09-26
Category : Mathematics
ISBN : 9783030488260

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Introduction to Soergel Bimodules by Ben Elias,Shotaro Makisumi,Ulrich Thiel,Geordie Williamson Pdf

This book provides a comprehensive introduction to Soergel bimodules. First introduced by Wolfgang Soergel in the early 1990s, they have since become a powerful tool in geometric representation theory. On the one hand, these bimodules are fairly elementary objects and explicit calculations are possible. On the other, they have deep connections to Lie theory and geometry. Taking these two aspects together, they offer a wonderful primer on geometric representation theory. In this book the reader is introduced to the theory through a series of lectures, which range from the basics, all the way to the latest frontiers of research. This book serves both as an introduction and as a reference guide to the theory of Soergel bimodules. Thus it is intended for anyone who wants to learn about this exciting field, from graduate students to experienced researchers.

Author : Paul B. Garrett
Publisher : CRC Press
Page : 396 pages
File Size : 50,6 Mb
Release : 1997-04-01
Category : Mathematics
ISBN : 041206331X

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Buildings and Classical Groups by Paul B. Garrett Pdf

Buildings are highly structured, geometric objects, primarily used in the finer study of the groups that act upon them. In Buildings and Classical Groups, the author develops the basic theory of buildings and BN-pairs, with a focus on the results needed to apply it to the representation theory of p-adic groups. In particular, he addresses spherical and affine buildings, and the "spherical building at infinity" attached to an affine building. He also covers in detail many otherwise apocryphal results. Classical matrix groups play a prominent role in this study, not only as vehicles to illustrate general results but as primary objects of interest. The author introduces and completely develops terminology and results relevant to classical groups. He also emphasizes the importance of the reflection, or Coxeter groups and develops from scratch everything about reflection groups needed for this study of buildings. In addressing the more elementary spherical constructions, the background pertaining to classical groups includes basic results about quadratic forms, alternating forms, and hermitian forms on vector spaces, plus a description of parabolic subgroups as stabilizers of flags of subspaces. The text then moves on to a detailed study of the subtler, less commonly treated affine case, where the background concerns p-adic numbers, more general discrete valuation rings, and lattices in vector spaces over ultrametric fields. Buildings and Classical Groups provides essential background material for specialists in several fields, particularly mathematicians interested in automorphic forms, representation theory, p-adic groups, number theory, algebraic groups, and Lie theory. No other available source provides such a complete and detailed treatment.

Author : Richard Kane
Publisher : Springer Science & Business Media
Page : 664 pages
File Size : 51,6 Mb
Release : 2001-06-21
Category : Mathematics
ISBN : 038798979X

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Reflection Groups and Invariant Theory by Richard Kane Pdf

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.

Author : Howard Hiller
Publisher : Pitman Publishing
Page : 230 pages
File Size : 53,6 Mb
Release : 1982
Category : Mathematics
ISBN : UOM:39015049314548

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Geometry of Coxeter Groups by Howard Hiller Pdf

Author : Marcelo Aguiar
Publisher : American Mathematical Soc.
Page : 201 pages
File Size : 40,5 Mb
Release : 2006
Category : Education
ISBN : 9780821853542

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Coxeter Groups and Hopf Algebras by Marcelo Aguiar Pdf

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.

Author : Nalini Joshi
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 51,6 Mb
Release : 2019-05-30
Category : Differential equations, Nonlinear
ISBN : 9781470450380

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Discrete Painlevé Equations by Nalini Joshi Pdf

Discrete Painlevé equations are nonlinear difference equations, which arise from translations on crystallographic lattices. The deceptive simplicity of this statement hides immensely rich mathematical properties, connecting dynamical systems, algebraic geometry, Coxeter groups, topology, special functions theory, and mathematical physics. This book necessarily starts with introductory material to give the reader an accessible entry point to this vast subject matter. It is based on lectures that the author presented as principal lecturer at a Conference Board of Mathematical Sciences and National Science Foundation conference in Texas in 2016. Instead of technical theorems or complete proofs, the book relies on providing essential points of many arguments through explicit examples, with the hope that they will be useful for applied mathematicians and physicists.