Mirrors and Reflections (eBook)

- 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.