Finite Reflection Groups (Graduate Texts in Mathematics (99)) 2nd Edition by L.C. Grove (PDF)

Description

Chapter 1 introduces some of the terminology and notation used later and indicates prerequisites. Chapter 2 gives a reasonably thorough account of all finite subgroups of the orthogonal groups in two and three dimensions. The presentation is somewhat less formal than in succeeding chapters. For instance, the existence of the icosahedron is accepted as an empirical fact, and no formal proof of existence is included. Throughout most of Chapter 2 we do not distinguish between groups that are “geo­ metrically indistinguishable,” that is, conjugate in the orthogonal group. Very little of the material in Chapter 2 is actually required for the sub­ sequent chapters, but it serves two important purposes: It aids in the development of geometrical insight, and it serves as a source of illustrative examples. There is a discussion offundamental regions in Chapter 3. Chapter 4 provides a correspondence between fundamental reflections and funda­ mental regions via a discussion of root systems. The actual classification and construction of finite reflection groups takes place in Chapter 5. where we have in part followed the methods of E. Witt and B. L. van der Waerden. Generators and relations for finite reflection groups are discussed in Chapter 6. There are historical remarks and suggestions for further reading in a Post lude.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book develops some theory of the Coxeter groups and is quite well-suited for self study. The content of the book is as follows: The first chapter reviews the basic facts from linear algebra (e.g. eigenvalues and Euclidean topology) and elementary group theory (e.g. orbit-stabilizer theorem) that are required to understand the first five chapters of the book. Chapter 2 grounds the reader with some geometrical insight in order to motivate the later material. Here the reader might have some trouble visualizing symmetry groups in three dimensions, but this is no cause for concern since this is not at all essential for understanding the rest of the book. Chapter 3 introduces fundamental domains and chapter 4 discusses root systems. For the uninitiated, this chapter may seem technical and unmotivated, though not especially difficult. One must simply take on faith that the 19 propositions/theorems here are leading somewhere interesting. So they do, in chapter 5, which is the heart of the book. Herein is contained the classical classification of the finite Coxeter groups, established by use of Coxeter graphs. The combinatorial methods in this chapter are quite elegant. Chapter 6 gives the alternative characterization of Coxeter groups in terms of generators and relations; this is the only part of the book that requires a basic understanding of free groups. Chapter 7 concludes with a study of the polynomial invariants of the Coxeter groups and the Shephard-Todd Theorem.For the interested undergraduate who has a grounding in the (modest) prerequisite background, this book will serve as a gentle and quite readable introduction to Coxeter groups. It will also be of use to the graduate student who should be able to read most of the book in a very short time.The only complaint to be made is that there is no mention of the representation theory of Coxeter groups. However, this is probably missing since the author wished to keep prerequisites to a minimum, and in any case, after reading this book one is well-prepared to read some of the more advanced topics elsewhere.

⭐This is not my review; but I have consciously read this book (chapters 1,2,3,4,5) for preparing my thesis, and I was thinking about the translation (from English to Spanish)of this book.

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