Finite Reflection Groups

Author: L.C. Grove
Publisher: Springer Science & Business Media
ISBN: 9781475718690
Release Date: 2013-03-09
Genre: Mathematics

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 Coxeter Groups

Author: James E. Humphreys
Publisher: Cambridge University Press
ISBN: 0521436133
Release Date: 1992-10-01
Genre: Mathematics

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.

Mirrors and Reflections

Author: Alexandre V. Borovik
Publisher: Springer Science & Business Media
ISBN: 9780387790664
Release Date: 2009-11-07
Genre: Mathematics

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.

Reflection Groups and Invariant Theory

Author: Richard Kane
Publisher: Springer Science & Business Media
ISBN: 9781475735420
Release Date: 2013-03-09
Genre: Mathematics

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.

Vorlesungen ber die Theorie der algebraischen Zahlen

Author: Erich Hecke
Publisher: University of Pennsylvania Press
ISBN: 0821821431
Release Date: 2000
Genre: Mathematics

This title has been described as An elegant and comprehensive account of the modern theory of algebraic numbers - Bulletin of the AMS.

Introduction to Complex Reflection Groups and Their Braid Groups

Author: Michel Broué
Publisher: Springer
ISBN: 9783642111754
Release Date: 2010-01-28
Genre: Mathematics

This book covers basic properties of complex reflection groups, such as characterization, Steinberg theorem, Gutkin-Opdam matrices, Solomon theorem and applications, including the basic findings of Springer theory on eigenspaces.

Coxeter Matroids

Author: Alexandre V. Borovik
Publisher: Springer Science & Business Media
ISBN: 9781461220664
Release Date: 2012-12-06
Genre: Mathematics

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.

The Finite Simple Groups

Author: Robert Wilson
Publisher: Springer Science & Business Media
ISBN: 9781848009875
Release Date: 2009-12-14
Genre: Mathematics

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].

Lie Algebras and Algebraic Groups

Author: Patrice Tauvel
Publisher: Springer Science & Business Media
ISBN: 9783540274278
Release Date: 2006-03-30
Genre: Mathematics

Devoted to the theory of Lie algebras and algebraic groups, this book includes a large amount of commutative algebra and algebraic geometry so as to make it as self-contained as possible. The aim of the book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included, and some recent results are discussed in the final chapters.


Author: Kenneth S. Brown
Publisher: Springer Science & Business Media
ISBN: 9781461210191
Release Date: 2013-06-29
Genre: Mathematics

For years I have heard about buildings and their applications to group theory. I finally decided to try to learn something about the subject by teaching a graduate course on it at Cornell University in Spring 1987. This book is based on the not es from that course. The course started from scratch and proceeded at a leisurely pace. The book therefore does not get very far. Indeed, the definition of the term "building" doesn't even appear until Chapter IV. My hope, however, is that the book gets far enough to enable the reader to tadle the literat ure on buildings, some of which can seem very forbidding. Most of the results in this book are due to J. Tits, who originated the the ory of buildings. The main exceptions are Chapter I (which presents some classical material), Chapter VI (which prcsents joint work of F. Bruhat and Tits), and Chapter VII (which surveys some applications, due to var ious people). It has been a pleasure studying Tits's work; I only hope my exposition does it justice.

Unitary Reflection Groups

Author: Gustav I. Lehrer
Publisher: Cambridge University Press
ISBN: 9780521749893
Release Date: 2009-08-13
Genre: Mathematics

A complex reflection is a linear transformation which fixes each point in a hyperplane. Intuitively, it resembles the transformation an image undergoes when it is viewed through a kaleidoscope, or arrangement of mirrors. This book gives a complete classification of all groups of transformations of n-dimensional complex space which are generated by complex reflections, using the method of line systems. In particular: irreducible groups are studied in detail, and are identified with finite linear groups; reflection subgroups of reflection groups are completely classified; the theory of eigenspaces of elements of reflection groups is discussed fully; 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.

Algebra 1

Author: Bartel L. van der Waerden
Publisher: Springer-Verlag
ISBN: 9783662015131
Release Date: 2013-04-17
Genre: Mathematics

Ziel des Buches. Die "abstrakte", "formale" oder "axiomatische" Richtung, der die Algebra ihren erneuten Aufschwung verdankt, hat vor allein in der Gruppentheorie, der Körpertheorie, der Bewertun- theorie, der Idealtheorie und der Theorie der hyperkomplexen Zahlen zu einer Reihe von neuartigen Begriffsbildungen, zur Einsicht in neue Zusammenhänge und zu weitreichenden Resultaten geführt. In diese ganze Begriffswelt den Leser einzuführen, soll das Hauptziel dieses Buches sein. Stehen demnach allgemeine Begriffe und Methoden im Vordergrun4, so sollen doch auch die Einzelresultate, die zum klassischen Bestand der Algebra gerechnet werden müssen. eine gehörige Berücksichtigung im Rahmen des modernen Aufbaus finden. Einteilung. Anweisungen für die Leser. Um die allgemeinen Gesichtspunkte, welche die "abstrakte" Auffassung der Algebra be herrschen, genügend klar zu entwickeln, war es notwendig, die Grund lagen der Gruppentheorie und der elementaren Algebra von Anfang an neu darzustellen. Angesichts der vielen in neuerer Zeit erschien~Ilen guten Darstellungen der Gruppentheorie, der klassischen Algebra und der Körpertheorie ergab sich die Möglichkeit, diese einleitenden Teile knapp (aber lückenlos) zu fassen. Eine breitere Darstellungkann der Anfänger jetzt überall finden!. Als weiteres Leitprinzip diente die Forderung, daß möglichst jeder einzelne Teil für sich allein verständlich sein soll. Wer die allgemeine Idealtheorie oder die Theorie der hyperkomplexen Zahlen kennenlernen will, braucht nicht die GALOIssche Theorie vorher zu studieren, und umgekehrt; und wer etwas über Elimination oder lineare Algebra nach schlagen will, darf nicht durch komplizierte idealtheoretische Begriffs bildungen abgeschreckt werden.