Notice bibliographique
- Notice
Type(s) de contenu et mode(s) de consultation : Texte noté : électronique
Auteur(s) : Borovik, Alexandre
Titre(s) : Mirrors and reflections [Texte électronique] : the geometry of finite reflection groups / Alexandre V. Borovik, Anna Borovik
Publication : New York : Springer, cop. 2010
Description matérielle : 1 ressource dématérialisée
Collection : Universitext
Note(s) : Includes bibliographical references and index
"Mirrors and Reflections presents an intuitive and elementary introduction to finite
reflection groups. Starting with basic principles, this book provides a comprehensive
classification of the various types of finite reflection groups and describes their
underlying geometric properties. Unique to this text is its emphasis on the intuitive
geometric aspects of the theory of reflection groups, making the subject more accessible
to the novice. Primarily self-contained, necessary geometric concepts are introduced
and explained. Principally designed for coursework, this book is saturated with exercises
and examples of varying degrees of difficulty. An appendix offers hints for solving
the most difficult problems. Wherever possible, concepts are presented with pictures
and diagrams intentionally drawn for easy reproduction. Finite reflection groups is
a topic of great interest to many pure and applied mathematicians. Often considered
a cornerstone of modern algebra and geometry, an understanding of finite reflection
groups is of great value to students of pure or applied mathematics. Requiring only
a modest knowledge of linear algebra and group theory, this book is intended for teachers
and students of mathematics at the advanced undergraduate and graduate levels."--Back
cover of book
Autre(s) auteur(s) : Borovik, Anna. Fonction indéterminée
Sujet(s) : Groupes de Coxeter
Groupes de réflexions (mathématiques)
Indice(s) Dewey :
512.23 (23e éd.) = Groupes finis (mathématiques)
Identifiants, prix et caractéristiques : ISBN 9780387790664
Identifiant de la notice : ark:/12148/cb446441155
Notice n° :
FRBNF44644115
(notice reprise d'un réservoir extérieur)
Table des matières : Part I. Geometric background ; 1. Affine Euclidean Space ARn ; 1.1. Euclidean Space
Rn ; 1.2. Affine Euclidean Space ARn ; 1.3. Affine Subspaces ; 1.4. Half-Spaces
; 1.5. Bases and Coordinates ; 1.6. Convex Sets ; 2. Isometries of ARn ; 2.1.
Fixed Points of Groups of Isometries ; 2.2. Structure of Isom ARn ; 3. Hyperplane
Arrangements ; 3.1. Faces of a Hyperplane Arrangement ; 3.2. Chambers ; 3.3. Galleries
; 3.4. Polyhedra ; 4. Polyhedral Cones ; 4.1. Finitely Generated Cones ; 4.2.
Simple Systems of Generators ; 4.3. Duality ; 4.4. Duality for Simplicial Cones
; 4.5. Faces of a Simplicial Cone ; Part II. Mirrors, Reflections, Roots ; 5. Mirrors
and Reflections ; 6. Systems of Mirrors ; 6.1. Systems of Mirrors ; 6.2. Finite
Reflection Groups ; 7. Dihedral Groups ; 7.1 Groups Generated by Two Involutions
; 7.2. Proof of Theorem 7.1 ; 7.3. Dihedral Groups: Geometric Interpretation ;
8. Root Systems ; 8.1. Mirrors and Their Normal Vectors ; 8.2. Root S
9. Root Systems An-1, BCn, Dn ; 9.1. Root System An-1 ; 9.2. Root Systems of Types
Cn and Bn ; 9.3. The Root System Dn ; Part III. Coxeter Complexes ; 10. Chambers
; 11. Generation ; 11.1. Simple Reflections ; 11.2. Foldings ; 11.3. Galleries
and Paths ; 11.4. Action of W on C ; 11.5 Paths and Foldings ; 11.6. Simple Transitivity
of W on C: Proof of Theorem 11.6 ; 12. Coxeter Complex ; 12.1. Labeling of the Coxeter
Complex ; 12.2. Length of Elements in W ; 12.3. Opposite Chamber ; 12.4. Isotropy
Groups ; 12.5. Parabolic Subgroups ; 13. Residues ; 13.1. Residues ; 13.2. Example
; 13.3. The Mirror System of a Residue ; 13.4. Residues are Convex ; 13.5. Residues:
the Gate property ; 13.6. The Opposite Chamber ; 14. Generalized Permutahedra ;
Part IV. Classification ; 15. Generators and Relations ; 15.1. Reflection Groups
are Coxeter Groups ; 15.2. Proof of Theorem 15.1 ; 16. Classification of Finite
Reflection Groups ; 16.1. Coxeter Graph ; 16.2. Decomp
