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Mirrors and Reflections: The Geometry of Finite Reflection Groups

A. V. Borovik and Anna Borovik
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Gizem Karaali
, on

The theory of finite reflection groups is a beautiful story which can be told at various levels. Many mathematicians get there only after a path that goes through Lie theory and representation theory. Many times the longer path is worth it, if only for the intuition it provides. There are many other roses to smell along the way, so taking a longer path is definitely not to be frowned upon. Nonetheless, finite reflection groups can also be introduced to students who have only a working knowledge of linear algebra, and the algebraic, the geometric and the combinatorial can be combined to create a capstone experience for undergraduates which will fulfill the needs and wishes of most. This book is a masterful exposition that does just that.

The book starts off with the premise that the best way to describe the theory of finite reflection groups is through their intrinsic geometry. Thus the book emphasizes hyperplane arrangements first and foremost. These are introduced very intuitively, in the form of mirror arrangements which create kaleidoscopic images. This recurring theme of mirrors and kaleidoscopes makes finite reflection groups real and concrete.

The focus is decidedly on the geometric intution. Readers do not need to know much group theory, though some group-theoretic concepts and results are used every now and then. Nonetheless, except for one instance in Section 7.3 where the formula for the length of a group orbit is used, the algebraic requirements expected of the reader are quite minimal.

There are many pictures scattered across the pages of the book, and these make the text much easier to follow. There is a brief discussion at the end on “The Forgotten Art of Blackboard Drawing,” where the authors express their opinion that pictures should accompany mathematical text and teaching, and that the drawings created should be reproducible by the students and the teachers alike. Thus, most of the figures in the book are simple illustrations. It would take some practice and experience to draw some of the more complex figures, such as Fig.14.1 which shows a permutahedron associated to the symmetric group on four letters, or Fig.19.1, which shows a tessellation of a sphere associated to the icosahedron. However the philosophy of the authors is quite evident, both from its brief description in the appendix and throughout, via the use of many many figures.

There are many good books on finite reflection groups out there. Among these the book by Humphreys, Reflection Groups and Coxeter Groups, is a classic. However Humphreys requires a lot from his readers; the intended reader is at least a graduate student. Furthermore, most relevantly to the review here, geometry is secondary to the whole discussion. Humphreys is most interested in how the algebra and the combinatorics of the theory are entangled, the text does not have many illustrations. The book by Grove and Benson, Finite Reflection Groups, is perhaps better suited to an undergraduate audience. It is an excellent text as well, but its focus is also mainly algebraic. Then there is Bourbaki, which is of course a classic, but it is a rare undergraduate who will get much, if anything, out of it. Graduate students and researchers may read and cite it, but I doubt any sane person would recommend it for undergraduate classroom use.

Borovik and Borovik is a welcome addition to this company of good books. It is certainly the most accessible, largely due to its geometric approach. Any algebraic discussion is preceded and followed up by some geometry and intuition connecting the discussion to an arrangement of mirrors. The material covered is extensive, and at the end of the book students will be well-prepared to follow some of the recent research in combinatorial representation theory. Concepts like paths, folding, galleries, etc. do not make it to most of the other books so early on, if they show up at all. However they are intuitively obvious notions once one follows the Borovik in their visual-rich excursion.

The book is appropriate both for classroom use and for outside reading. There are exercises after each of the twenty sections and there are answers or hints at the end of the book for most of these. The questions range from the standard computational to the more theoretical to the all-together common-sensical. For instance Exercise 6.1 asks a question all children ask their parents at least once: “why is it that a mirror changes left and right but does not change up and down?” (The answer is in the back of the book!)

I highly recommend this book to instructors looking for a good text for a capstone course, and for students, both undergraduate and graduate, looking to learn about finite reflection groups. If they prefer, they may follow it up with some of the more advanced texts, but they will have a solid foundation of theory and intuition once they are done with this book.

Gizem Karaali is assistant professor of mathematics at Pomona College. She came to finite reflection groups through the longer winding path, and enjoyed the view all along the way.


- 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.