Shattered Symmetry. Group Theory From the Eightfold Way to the Periodic Table . By Pieter Thyssen and Arnout Ceulemans. Oxford University Press, 2017. Pp. 528. Price GBP 52.00 (hardback). ISBN 9780190611392.

The subtitle of this marvellous book could have been: ‘All you always wanted to know about Lie groups, but were too afraid to ask’. Most of the time, books dealing with such a subject propose a mathematical treatment that goes well beyond the basic calculus knowledge of physicists and chemists. This one uses simple calculus and takes the reader by the hand through all chapters in a step-by-step way; difficulties are handled gradually in a very pedagogical presentation. It starts with the example of usual groups – usual in the sense that they are known by every chemist or physicist – and ends with the proposition of a very complex Lie group [SO(4, 2)] to explain, or at least to rationalize, the structure of the periodic table and the (n + l) rule. Every chapter provides the reader with a snippet of new knowledge, in order to allow her or him to appreciate and understand each landscape at the heart of the Lie-group world. The reader will also find a collection of historical anecdotes and illustrations based on the journey of Alice when she goes through the mirror, just as the reader travels across the wonderland of Lie groups and their implications in physics and chemistry. As may be apparent, I really appreciated reading this book and I recommend it to anyone willing to understand this powerful tool. Below, this review comments on each of the 14 chapters and the 12 appendices in order of appearance. The first chapter is dedicated to the definition of several notions needed to grasp the meaning of the remaining parts of the book. It is based on the point group of the triangle that everyone can understand. The inspiring life of Evarist Gallois is detailed to give historical markers. With the second chapter one takes a step through the mirror with the elements of group theory and the concept of symmetry breaking, isomorphisms and homomorphisms. Again, usual groups are used to guide the reader and an historical interlude concludes the chapter. Armed with definitions from the first two chapters, the reader can take a step further into the world of continuous groups with the study of the axial rotation group which is the subject of the third chapter. The matrix representations are discovered and used to build the O(2) and SO(2) groups. The latter is used throughout the fourth chapter to introduce the infinitesimal generators that are going to be used in the following chapters. The system of planar ring molecules, which is well known by chemists, is used to illustrate the power of group theory and in particular the group SO(2). The fifth chapter adds some complexity to the study in going from rotation in a plane to rotation in three dimensions. The SO(3) and O(3) groups are built and most importantly the Lie algebra is presented together with Casimir invariants. The role of these invariants in quantum mechanics is tremendous. In a similar way to the previous chapter, the fifth one ends with the study of the system of buckminsterfullerene. All features presented in the first five chapters are mathematically grounded in the scholium that forms Chapter 6. Here the three steps connecting the Lie algebra of a Lie group to the spectrum of the Hamiltonian are detailed going from the Cartan subalgebra and Cartan generators to Casimir invariants via Weyl generators and Weyl diagrams. It is certainly the most important chapter and it ends the first part of the book, focused on space symmetries. ISSN 2053-2733