Reflection Groups and Invariant Theory
The main theme of this book is a study of the properties of reflection groups (finite groups generated by reflections in a Euclidean space), their generalisations and invariant theory connected with them. Suitable generalisations of reflections in Euclidean spaces are pseudo-reflections in a vector space V: these are linear maps f from V to V of the form f(u) = u – (a*, u)a, with a* in V* and a in V. Groups generated by pseudo-refelections are called pseudo-reflection groups.
The book is in three parts. The first part contains a study of reflection groups and their associated Coxeter systems, using root systems as a tool: Weyl groups and their crystallographic root systems form an important special case. The second important tool used in the book is invariant theory. A natural framework for a study of invariant theory is the concept of pseudo-reflection group, and this theme is carefully investigated in the second part. The final part contains a discussion of conjugacy classes of elements and subgroups for reflection groups (such as a relation of Coxeter elements to the underlying root system), as well as a use of invariant theory in the study of eigenvalues of elements from pseudo-reflection groups.
The topics treated in the book play a key role in other important branches of mathematics (Lie groups and Lie algebras and their applications in geometry and analysis or theory of algebraic groups). The book is nicely organised and written in a very understandable way. The main ideas are clearly explained at the beginning of each chapter or section, so it is easy to learn the main facts first and to fill in the details later, when needed. For the main part of the book, only a basic knowledge of linear algebra and algebra is needed, and short summaries of more advanced knowledge are enclosed for the convenience of the reader. The book is pleasant to read, and can be heartily recommended to a general mathematical audience, starting with graduate students.