Mirrors and Reflections: The Geometry of Finite and Affine 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. Numerous exercises at various levels of difficulty are included.

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.

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.

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 fur Mathematik, Vol. 162 (2), February, 2011)

Preface		v
Part I Geometric Background
Affine Euclidean Space ARn		3	(8)
Euclidean Space Rn		4	(1)
Affine Euclidean Space ARn		5	(1)
Affine Subspaces		6	(2)
Subspaces		6	(1)
Systems of Linear Equations		6	(1)
Points and Lines		6	(1)
Planes		7	(1)
Hyperplanes		7	(1)
Orthogonal Projection		7	(1)
Half-Spaces		8	(1)
Bases and Coordinates		8	(1)
Convex Sets		9	(2)
Isometriesof ARn		11	(6)
Fixed Points of Groups of Isometries		11	(1)
Structure of Isom ARn		12	(5)
Translations		12	(1)
Orthogonal Transformations		13	(4)
Hyperplane Arrangements		17	(8)
Faces of a Hyperplane Arrangement		17	(1)
Chambers		18	(1)
Galleries		19	(1)
Polyhedra		20	(5)
Polyhedral Cones		25	(12)
Finitely Generated Cones		25	(2)
Cones		25	(1)
Extreme Vectors and Edges		26	(1)
Simple Systems of Generators		27	(2)
Duality		29	(1)
Duality for Simplicial Cones		30	(1)
Faces of a Simplicial Cone		31	(6)
Part II Mirrors, Reflections, Roots
Mirrors and Reflections		37	(4)
Systems of Mirrors		41	(8)
Systems of Mirrors		41	(3)
Finite Reflection Groups		44	(5)
Dihedral Groups		49	(6)
Groups Generated by Two Involutions		49	(1)
Proof of Theorem 7.1		50	(1)
Dihedral Groups: Geometric Interpretation		51	(4)
Root Systems		55	(8)
Mirrors and Their Normal Vectors		55	(1)
Root Systems		56	(1)
Planar Root Systems		57	(2)
Positive and Simple Systems		59	(4)
Root Systems An---1, BCn, Dn		63	(16)
Root System An-1		63	(5)
A Few Words about Permutations		63	(1)
Permutation Representation of Symn		64	(1)
Regular Simplices		64	(1)
The Root System An-1		65	(1)
The Standard Simple System		66	(1)
Action of Symn on the Set of all Simple Systems		66	(2)
Root Systems of Types Cn and Bn		68	(4)
Hyperoctahedral Group		68	(1)
Admissible Orderings		69	(1)
Root Systems Cn and Bn		70	(1)
Action of W on &Phis;		71	(1)
The Root System Dn		72	(7)
Part III Coxeter Complexes
Chambers		79	(4)
Generation		83	(8)
Simple Reflections		83	(1)
Foldings		84	(1)
Galleries and Paths		85	(2)
Action of W on C		87	(1)
Paths and Foldings		87	(2)
Simple Transitivity of W on C: Proof of Theorem 11.6		89	(2)
Coxeter Complex		91	(8)
Labeling of the Coxeter Complex		91	(2)
Length of Elements in W		93	(1)
Opposite Chamber		93	(1)
Isotropy Groups		94	(1)
Parabolic Subgroups		95	(4)
Residues		99	(6)
Residues		99	(1)
Example		100	(1)
The Mirror System of a Residue		101	(1)
Residues are Convex		102	(1)
Residues: the Gate Property		102	(1)
The Opposite Chamber		103	(2)
Generalized Permutahedra		105	(8)
Part IV Classification
Generators and Relations		113	(4)
Reflection Groups are Coxeter Groups		113	(2)
Proof of Theorem 15.1		115	(2)
Classification of Finite Reflection Groups		117	(6)
Coxeter Graph		117	(1)
Decomposable Reflection Groups		118	(1)
Labeled Graphs and Associated Bilinear Forms		118	(1)
Classification of Positive Definite Graphs		119	(4)
Construction of Root Systems		123	(10)
Root System An		123	(1)
Root System Bn, n ≥ 2		124	(1)
Root System Cn, n ≥ 2		125	(1)
Root System Dn, n ≥ 4		126	(1)
Root System E8		126	(1)
Root System E7		127	(1)
Root System E6		128	(1)
Root System F4		128	(1)
Root System G2		129	(1)
Crystallographic Condition		129	(4)
Orders of Reflection Groups		133	(6)
Part V Three-Dimensional Reflection Groups
Reflection Groups in Three Dimensions		139	(8)
Planar Mirror Systems		139	(1)
From Mirror Systems to Tessellations of the Sphere		139	(2)
The Area of a Spherical Triangle		141	(1)
Classification of Finite Reflection Groups in Three Dimensions		142	(5)
Icosahedron		147	(20)
Construction		147	(2)
Uniqueness and Rigidity		149	(2)
The Symmetry Group of the Icosahedron		151	(6)
Part VI Appendices
A The Forgotten Art of Blackboard Drawing		157	(4)
B Hints and Solutions to Selected Exercises		161	(6)
References		167	(2)
Index		169