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Unitary Reflection Groups

Gustav I. Lehrer and Donald E. Taylor
Cambridge University Press
Publication Date: 
Number of Pages: 
Australian Mathematical Society Lecture Series 20
[Reviewed by
Gizem Karaali
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Several expositions of the theory of finite reflection groups exist to satisfy readers of all stripes. The analogous theory of complex reflection groups, on the other hand, did not enjoy a uniform treatment until this book.

The book under review is an exposition of the full classification of complex reflection groups, those groups generated by complex reflections, i.e., complex linear transformations of finite order which fix pointwise a unique hyperplane. The classification is due to G. C. Shephard and J. A. Todd (Canadian J. Math. 6 (1954), 274–304). It involves three infinite families as well as 34 exceptional cases. Also provided in the book are some developments of the invariant theory of complex reflection groups, and of reflection cosets. An appendix describes many connections to other fields of mathematics for those who might be interested.

Even this oversimplified summary of the main theme of the book might provide some insight into its nature. The book is quite technical in many instances, and develops in detail all the necessary tools to obtain and understand the result, whose original proof depended on various earlier works from different threads of research.

In theory one can agree with the authors' assertion that the text could be used by graduate students who have a solid grasp of undergraduate algebra, but my choice for a more natural audience would be mathematicians who may have heard of complex reflection groups or come across them in their research and who wish to know more. The technical prerequisites are indeed limited to a solid grasp of basic algebra, but in order to enjoy and benefit most from the book a reader will most likely need to have some motivation to learn this material that goes beyond simply passing another course. It is beautiful mathematics, and it is mostly self-contained, but definitely not a light read for those who are looking for some fun mathematics to enjoy before bedtime.

Gizem Karaali is assistant professor of mathematics at Pomona College.

Introduction; 1. Preliminaries; 2. The groups G(m, p, n); 3. Polynomial invariants; 4. Poincaré series and characterisations of reflection groups; 5. Quaternions and the finite subgroups of SU2(C); 6. Finite unitary reflection groups of rank two; 7. Line systems; 8. The Shepherd and Todd classification; 9. The orbit map, harmonic polynomials and semi-invariants; 10. Covariants and related polynomial identities; 11. Eigenspace theory and reflection subquotients; 12. Reflection cosets and twisted invariant theory; A. Some background in commutative algebra; B. Forms over finite fields; C. Applications and further reading; D. Tables; Bibliography; Index of notation; Index.