Linear Representation of Groups

When I was a graduate student at UCSD in the early 1980s, the world of number theory had already for quite a while been aflame with the theory of modular forms. I knew about this from my undergraduate days “a couple of hours up the freeway” at UCLA, from work with V. S. Varadarajan and Basil Gordon. I had learned about the classical theory of elliptic modular forms from Gordon in his number theory seminar, and Varadarajan had introduced me to some deep themes in this important and beautiful subject both in a seminar using, among other sources, Introduction to the Arithmetic Theory of Automorphic Forms by Gôro Shimura, and in a reading course centered on André Weil’s superb monograph, Elliptic Functions According to Eisenstein and Kronecker.

So when Audrey Terras, my graduate advisor, showed me a list of eight thesis problems to work on, I picked the last one on the list because it had to do with other work by Weil, specifically, his 1966 paper on what is sometimes (somewhat euphemistically) referred to as “the converse theorem” to the Hecke correspondence. I soon realized that one of the things most conspicuous by its absence in my preparation for my thesis work was representation theory, and so I scurried about in something of a mad frenzy, learning what I needed “on the fly” from a hodgepodge of sources, primary among them a superb LMS offering titled Representation Theory of Lie Groups

Well, it all turned out fine (I finished my thesis), but I came away from the whole business with a somewhat eccentric knowledge of representation theory. For one thing my focus in my thesis was primarily on unitary groups and algebraic groups, and not on finite groups, so I was sorely deficient in what is generally presented as a preliminary theme, and the locale for accessible computations, all well before one gets to locally compact groups (for example). Some of these gaps needed to be plugged later, and were, at least to some degree (I am ashamed to admit that, even now, my knowledge of representation theory is variable) and over the years I have had occasion to study parts of, e.g., Kirillov’s wonderful book on the subject, Weil’s own l’Intégration dans les Groupes Topologiques et ses Applications, and, of course, Curtis-Reiner and Serre’s Linear Representations of Finite Groups. So I am not complaining at all …

However, in another order of Providence, I would have been ecstatic to have had Ernest B. Vinberg’s Linear Representations of Groups at my disposal in my student days: I would have done things right. (On the other hand, I was pretty goofy and headstrong as a graduate student so it might not have made as deep a dent as it should have. Youth, indeed, is wasted on the young.)

In a compact orbit of less than 150 pages, so to speak, Vinberg covers the basic theory of representations of finite, compact, and Lie groups, the author’s aim being “to give as simple and detailed an account as possible of the problems considered.” He adds the disclaimer: “The book therefore makes no claim to completeness [and] can in no way give a representative picture of the modern [situation]”; for the latter her refers to Kirillov’s acknowledged 1972 classic, Elements of the Theory of Representations. The date of this reference underscores the fact that the book under review, as being an offering in the “Modern Birkhäser Classics” series, is somewhat dated: Vinberg was writing in the1980s, and the book under review is the English edition, appearing in 1989.

Thus, Vinberg’s book should be seen as preliminary to far more ambitious and weighty texts (just think of Curtis-Reiner; Kirillov’s book is also pretty beefy) and, as such, it certainly has not aged badly. It takes the reader from raw basics, done very nicely, through the obligatory discussion of the regular representation and orthogonality relations, to some excellent stuff on SU₂ and SO₃, with a discussion of Laplace’s spherical functions thrown in. The presence of a huge number of guiding examples is a major asset, of course, as is the large collection of exercises for the reader. It is indeed a marvelous introduction to representation theory!

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.