Mirrors and reflections. The geometry of finite reflection groups.
type: Item
instance of: scholarly article
title: Mirrors and reflections. The geometry of finite reflection groups.
zbMATH Open document ID: 1193.20001
DOI: 10.1007/978-0-387-79066-4
author: Alexandre V. Borovik
author: Anna Borovik
published in: Universitext
publication date: 2009-10-28T00:00:00Z
review text: 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 proper…
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 20-01
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 51-01
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 17B22
zbMATH DE Number: 5622438
zbMATH Keywords: reflection groups
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Coxeter complexes
zbMATH Keywords: reflections
zbMATH Keywords: hyperplane arrangements
zbMATH Keywords: affine Euclidean spaces
zbMATH Keywords: isometries
zbMATH Keywords: chambers
zbMATH Keywords: galleries
zbMATH Keywords: polyhedra
zbMATH Keywords: polyhedral cones
zbMATH Keywords: mirrors
zbMATH Keywords: root systems
zbMATH Keywords: generators and relations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-0-387-79066-4
OpenAlex ID: W2500263197
description: scientific article; zbMATH DE number 5622438