Mirrors and Reflections: The Geometry of Finite Reflection Groups

Paperback | November 10, 2009

byAlexandre V. Borovik, Anna Borovik

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Mirrors and Reflections is 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: . 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 . Few prerequisites are necessary beyond linear algebra and the basic notions of group theory.The exposition is directed at advanced undergraduates and first-year graduate students.

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From the Publisher

Mirrors and Reflections is 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: . Many important concepts in the proofs are illu...

From the Jacket

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 intuiti...

Format:PaperbackDimensions:184 pages, 9.25 × 6.1 × 0.27 inPublished:November 10, 2009Publisher:Springer New YorkLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0387790659

ISBN - 13:9780387790657

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Table of Contents

- Part I Geometric Background.- 1. Affine Euclidean Space ARn.-1.1 Euclidean Space Rn.- 1.2 Affine Euclidean Space ARn.- 1.3 Affine Subspaces.- 1.3.1 Subspaces.- 1.3.2 Systems of Linear Equations.- 1.3.3 Points and Lines .- 1.3.4 Planes .- 1.3.5 Hyperplanes.- 1.3.6 Orthogonal Projection.- 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 IsomARn .- 2.2.1 Translations.- 2.2.2 Orthogonal Transformations .- 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 .- 4.1.1 Cones.- .1.2 Extreme Vectors and Edges .- 4.2 Simple Systems of Generators.- 4.3 Duality .- 4.4 Duality for Simplicial Cones .- 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 Systems.- 8.3 Planar Root Systems.- 8.4 Positive and Simple Systems.- 9 Root Systems An¡1, BCn, Dn.- 9.1 Root System An¡1 .- 9.1.1 A Few Words about Permutations .- 9.1.2 Permutation Representation of Symn .- 9.1.3 Regular Simplices .- 9.1.4 The Root System An¡1 .- 9.1.5 The Standard Simple System.- 9.1.6 Action of Symn on the Set of all Simple Systems .- 9.2 Root Systems of Types Cn and Bn .- 9.2.1 Hyperoctahedral Group.- 9.2.2 Admissible Orderings.- 9.2.3 Root Systems Cn and Bn.- 9.2.4 Action of W on C.- 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 Decomposable Reflection Groups.- 16.3 Labeled Graphs and Associated Bilinear Forms.- 16.4 Classification of Positive Definite Graphs.- 17 Construction of Root Systems.- 17.1 Root System An.- 17.2 Root System Bn, n > 2.- 17.3 Root System Cn, n > 2.- 17.4 Root System Dn, n > 4.- 17.5 Root System E8.- 17.6 Root System E7 17.7 Root System E6.- 17.8 Root System F4 .- 9 Root System G2 .- 17.10 Crystallographic Condition .- 18 Orders of Reflection Groups .- Part V Three-Dimensional Reflection Groups.- 19 Reflection Groups in Three Dimensions.- 19.1 Planar Mirror Systems.- 19.2 From Mirror Systems to Tessellations of the Sphere.- 19.3 The Area of a Spherical Triangle.- 19.4 Classification of Finite Reflection Groups in Three Dimensions.- 20 Icosahedron.- 20.1 Construction.- 20.2 Uniqueness and Rigidity.- 20.3 The Symmetry Group of the Icosahedron.- Part VI Appendices.- A The Forgotten Art of Blackboard Drawing.- B Hints and Solutions to Selected Exercises.- References.- Index.

Editorial Reviews

From the reviews:"In Mirrors and Reflections by Alexandre Borovik (Univ. of Manchester, UK) and Anna Borovik, readers get the whole stage . . Mastering this book not only gives undergraduates a taste of the mathematics of special objects, but prepares the way to various more important abstract theories. Thoughtfully illustrated, compact, leisurely, and unique in its coverage, this work is the way to learn this critical material. Summing Up: Highly recommended. Academics students, all levels, and professionals." (D. V. Feldman, Choice, Vol. 48 (1), September, 2010)"A different approach to the study of reflection groups: an intuitive geometric approach, suitable for undergraduate students. . the book provides the reader with the necessary geometric background. . The book ends with a very interesting appendix on the 'forgotten art of blackboard drawing', where the authors give advice on making usable mathematical drawings. . the authors believe that pictures are indispensable tools which facilitate mathematical work. They also give hints and solutions to selected exercises." (Maria Chlouveraki, Mathematical Reviews, Issue 2011 b)"This is a nice booklet! . the authors present an almost purely geometric approach to the theory of reflection groups which can be followed even by undergraduates. . The authors attach great value to geometric intuition . which makes the theory easily accessible. A very recommendable booklet!" (G. Kowol, Monatshefte für Mathematik, Vol. 162 (2), February, 2011)