Mirrors and reflections: the geometry of finite reflection groups

Preface

Part I. Geometric Background

1. Affine Euclidean Space AR n

1.1. Euclidean Space R n

1.2. Affine Euclidean Space AR n

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 AR n

2.1. Fixed Points of Groups of Isometries

2.2. Structure of Isom AR n

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

4.1.2. Extreme Vectors and Edges

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 Systems

8.3. Planar Root Systems

8.4. Positive and Simple Systems

9. Root Systems A n-1 , BC n , D n

9.1. Root System A n-1

9.1.1. A Few Words about Permutations

9.1.2. Permutation Representation of Sym n

9.1.3. Regular Simplices

9.1.4. The Root System, A n-1

9.1.5. The Standard Simple System

9.1.6. Action of Sym n on the Set of all Simple Systems

9.2. Root Systems of Types C n and B n

9.2.1. Hyperoctahedral Group

9.2.2. Admissible Orderings

9.2.3. Root Systems C n and B n

9.2.4. Action of W on Φ

9.3. The Root System D n

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 A n

17.2. Root System B n , n ≥ 2

17.3. Root System C n , n ≥ 2

17.4. Root System D n , n ≥ 4

17.5. Root System E 8

17.6. Root System E 7

17.7. Root System E 6

17.8. Root System F 4

17.9. Root System G 2

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