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Author Borovik, Alexandre

Title Mirrors and reflections : the geometry of finite reflection groups / Alexandre V. Borovik, Anna Borovik

Publ Info New York : Springer, [2010]
2010
LOCATION CALL # STATUS NOTE
 RWU Main Library  QA177 .B67 2010    AVAILABLE
 URI  QA177 .B67 2010    AVAILABLE
Descript xii, 171 pages : illustrations ; 24 cm
text rdacontent
unmediated rdamedia
volume rdacarrier
Series Universitext
Note Includes bibliographical references (p. [167]) and index
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 -- 1.4. 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 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 -- 9.2. 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. 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 -- 17.9. 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
LC subject Reflection groups
Coxeter complexes
Add Author Borovik, Anna
ISBN 0387790659
9780387790657 (pbk.)
ISBN/ISSN 10.1007/978-0-387-79066-4