Reflection Groups And Coxeter Groups

Reflection Groups and Coxeter Groups PDF
Author: James E. Humphreys
Publisher: Cambridge University Press
Category : Mathematics
Languages : en
Pages : 204
View: 4885

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Book Description:
A self-contained graduate textbook introducing the basic theory of Coxeter groups.


Reflection Groups And Invariant Theory

Reflection Groups and Invariant Theory PDF
Author: Richard Kane
Publisher: Springer Science & Business Media
Category : Mathematics
Languages : en
Pages : 379
View: 4248

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Book Description:
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.


Finite Reflection Groups

Finite Reflection Groups PDF
Author: L.C. Grove
Publisher: Springer Science & Business Media
Category : Mathematics
Languages : en
Pages : 136
View: 594

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Book Description:
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.


Coxeter Matroids

Coxeter Matroids PDF
Author: Alexandre V. Borovik
Publisher: Springer Science & Business Media
Category : Mathematics
Languages : en
Pages : 266
View: 6890

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Book Description:
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.


Geometric Combinatorics

Geometric Combinatorics PDF
Author: Ezra Miller
Publisher: American Mathematical Soc.
Category : Mathematics
Languages : en
Pages : 691
View: 2679

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Book Description:
Geometric combinatorics describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. This text is a compilation of expository articles at the interface between combinatorics and geometry.


Characters Of Finite Coxeter Groups And Iwahori Hecke Algebras

Characters of Finite Coxeter Groups and Iwahori Hecke Algebras PDF
Author: Institut Girard Desargues Meinolf Geck
Publisher: Oxford University Press
Category : Mathematics
Languages : en
Pages : 446
View: 5841

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Book Description:
Finite Coxeter groups and related structures arise naturally in several branches of mathematics, for example, Lie algebras or theory of knots and links. This is the first book which develops the character theory of finite Coxeter groups and Iwahori-Hecke algebras in a systematic way, ranging from classical results to recent developments.


The Finite Simple Groups

The Finite Simple Groups PDF
Author: Robert Wilson
Publisher: Springer Science & Business Media
Category : Mathematics
Languages : en
Pages : 298
View: 7642

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Book Description:
Thisbookisintendedasanintroductiontoallthe?nitesimplegroups.During themonumentalstruggletoclassifythe?nitesimplegroups(andindeedsince), a huge amount of information about these groups has been accumulated. Conveyingthisinformationtothenextgenerationofstudentsandresearchers, not to mention those who might wish to apply this knowledge, has become a major challenge. With the publication of the two volumes by Aschbacher and Smith [12, 13] in 2004 we can reasonably regard the proof of the Classi?cation Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete. Thus it is timely to attempt an overview of all the (non-abelian) ?nite simple groups in one volume. For expository purposes it is convenient to divide them into four basic types, namely the alternating, classical, exceptional and sporadic groups. The study of alternating groups soon develops into the theory of per- tation groups, which is well served by the classic text of Wielandt [170]and more modern treatments such as the comprehensive introduction by Dixon and Mortimer [53] and more specialised texts such as that of Cameron [19].


Coxeter Groups And Hopf Algebras

Coxeter Groups and Hopf Algebras PDF
Author: Marcelo Aguiar
Publisher: American Mathematical Soc.
Category : Mathematics
Languages : en
Pages : 181
View: 4923

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Book Description:
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.


The Geometry And Topology Of Coxeter Groups Lms 32

The Geometry and Topology of Coxeter Groups   LMS 32  PDF
Author: Michael Davis
Publisher: Princeton University Press
Category : Mathematics
Languages : en
Pages : 600
View: 4433

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Book Description:
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.


Foundations Of Hyperbolic Manifolds

Foundations of Hyperbolic Manifolds PDF
Author: John Ratcliffe
Publisher: Springer Science & Business Media
Category : Mathematics
Languages : en
Pages : 782
View: 1042

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Book Description:
This heavily class-tested book is an exposition of the theoretical foundations of hyperbolic manifolds. It is a both a textbook and a reference. A basic knowledge of algebra and topology at the first year graduate level of an American university is assumed. The first part is concerned with hyperbolic geometry and discrete groups. The second part is devoted to the theory of hyperbolic manifolds. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds. Each chapter contains exercises and a section of historical remarks. A solutions manual is available separately.