Mirrors and reflections. The geometry of finite reflection groups.
The highlight in this book is a proof of the ``Classification of Finite Reflection Groups'' and the ``Construction of Root Systems''. In order to get there, the authors choose to develop, emphasize, and utilize geometric intuition. A set of reflections is identified with the arrangement of their reflecting hyperplanes (mirrors) in order to exploit their geometric structure. The geometry is emphasized throughout the book. There is an appendix on the ``Forgotten Art of Blackboard Drawing''. The ``Reflection Groups in Three Dimensions'' are explored by elementary spherical geometry. A great deal of knowledge that a student has acquired before taking a course on reflection groups, is applied to the topic at hand. The reader is supposed to have an intimate knowledge of linear algebra, in particular of the \(n\)-dimensional Euclidean space. Some of the necessary material is presented in the early chapters of the book, such as ``Fixed Points of Groups of Isometries'' or ``Orthogonal Transformations''. It is assumed that the reader has some knowledge in topology and in combinatorics. Hyperplane arrangements are presented early in the book where also chambers and galleries are introduced. The presentation touches on topics that are traditionally treated in linear programming. These may be applied to arrangements whose hyperplanes pass through a common point. Part II of this book is called ``Mirrors, Reflections, Roots''. Here more properties of finite reflection groups are presented, the dihedral groups are treated, a root system is defined as a finite set \(\Phi\) of vectors in \(V\) that satisfies \(\Phi\cap\mathbb{R}\rho=\{\rho,-\rho\}\) and \(s_\rho\Phi=\Phi\) for all \(\rho\in\Phi\), where \(s_\rho\) is the reflection along \(\rho\). The properties of positive roots and simple roots are studied. Finally, the root systems \(A_{n-1}\), \(B_n\), \(C_n\), and \(D_n\) are explored in detail and their geometric properties are described. In Part III, the group-theoretic properties of reflection groups, such as the reflection length of an element, are discussed and proved, using the material developed earlier, always observing that any reflection is determined by its reflecting hyperplane. In the process chambers and galleries come into play, more insight into the geometry structure is obtained. The numerous illustrations are an integral part of the presentation. In Part IV the reflection groups are classified and the root systems \(A_n\), \(B_n\), \(C_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\), \(F_4\), and \(G_2\) are constructed.
scientific article; zbMATH DE number 5622438