This book is a welcome addition to the classical subject of representation theory, which is now ubiquitous in mathematics. As there are celebrated introductory texts like the one by J.-P. Serre, a new book on the subject is expected to have a different viewpoint and should contain some recent results. The author as a contemporary number theorist brings a somewhat new perspective to the topic. In addition, the author brings some novel applications to number theory. An additional feature is a fusion of the functorial and the classical approaches. A case in point is the treatment of Burnside’s theorem. The various historical remarks through the text are very illuminating. For instance, the reviewer was unaware of the fact that the Jordan-Hölder theorem in the context of representation theory is due to Noether according to Weyl.
We describe the contents of the book chapter-wise now. The basic notions are discussed in chapter 2 which is a little more than 100 pages long. This forms the backbone of the text. The treatment is interspersed with a number of motivating and explanatory comments. This chapter is ideal for self-learning. In chapter 3, the author looks at representations in some analogous contexts like Lie algebras and group algebras and topological groups. The special case of finite groups is studied in chapter 4. A brief discussion of semi-simple rings and modules would have been a good idea. On the other hand, this chapter has some very interesting new applications to quasi-random groups and to relation spaces of irreducible polynomials. In addition, a number of enlightening remarks and examples are provided. For example, the theme that multiplicative relations between roots are rare if the Galois group is large, has been brought forth nicely. The main purpose of the fifth chapter is a proof of the Peter-Weyl theorem. The treatment is fairly detailed and contains also quite a few subsidiary remarks, examples and applications. For instance, it is shown how representation theory of \(SU(2)\) leads naturally to well-known properties of Chebychev polynomials. The application to representations of infinite products of compact groups is useful in number theory where adelic groups and their compact open subgroups arise naturally. The sixth chapter which details applications of representations of compact groups carries a very interesting new application known as the Larsen alternative. This alternative tells us that if the fourth moment of a compact subgroup of \(U(n)\) (which is always an integer) is \(2\), the subgroup must either be finite or \(SU(n)\). It has applications to hyper-Kloosterman sums some of which are discussed in this chapter. The chapter also has an analysis of the hydrogen atom via representation theory. As this chapter has various, interesting number-theoretic applications not detailed in any textbook so far, there is some licence to include a few more out-of-the-box topics which the author could have exploited more. For instance, the author could have added to the excitement by describing Gassmann equivalence of representations and written out applications to Dedekind zeta functions. A more general version of the Larsen alternative is given in the seventh chapter which comprises of a very brief study of the representations of algebraic groups. The case of \(SL(2,\mathbb R)\) is treated in complete detail here. Again, a reference to Weyl’s unitarian trick in this chapter would have been apt and would go well with the abundance of remarks made throughout the text.
The book contains several novel sub-topics. It would have been nice if the author had also discussed certain old results like Frobenius’s theorem on the number of solutions of \(x^n=1\) which is a striking application of basic character theory. Also, after Burnside’s \(p^aq^b\)-theorem was discussed, it would have been good to prove Philip Hall’s theorem on solvability. Further, Mackey’s irreducibility criterion could have been mentioned. A description of Weyl’s unitarian trick would have been motivating as well. Of course, these topics are discussed in older texts but including them in this book would have added to its value which is already considerable. An additional help would have been to point out which exercises are harder.
In conclusion, this text is comprehensive in its treatment of the basics and has many novel applications especially of interest in number theory. It is carefully written, has plenty of exercises to help learn the subject and, is a pleasure to read.