This charming book “arose from the lecture notes of a representation theory course given by [Pavel Etingof] to the remaining six authors in March 2004 within the framework of the Clay Mathematics Research Academy for high school students and its extended version given by [Etingof] to MIT undergraduate mathematics students in the fall of 2008.” Accordingly, the material in the book is quite modern, the unifying theme being the representation theory of associative algebras, and the presentation is very accessible, “to students with a strong background in linear algebra and a basic knowledge of abstract algebra.”
There are nine chapters in the book, or eight, actually, given that the Introduction is defined as the first chapter. After that the sequence is quite natural. It all starts with the basics, including a discussion of “the objects we will study … [i.e.] associative algebras, groups, quivers, and Lie algebras,” and a treatment of representations of associative algebras. Thereafter the focus falls on finite groups, covering two chapters (as is only right); naturally Frobenius and Burnside are featured here. After this we get to quivers, a pair of interludes on category theory and homological algebra, and (in the last chapter) “a short introduction to the representation theory of finite dimensional algebras.”
While the reader, obviously presumed to be a novice, will come away with a broad understanding and representative view of this subject of central importance to mathematics, he will have to proceed to other texts for a more in-depth treatment of the material. Accordingly the authors refer to some mainstays in this area, including Curtis and Reiner’s Representation Theory of Finite Groups and Associative Algebras and of course Serre’s beautiful Linear Representations of Finite Groups.
However, within the parameters the book under review set itself, it is a truly wonderful achievement. Introduction to Representation Theory is well-written, sportingly paced, and brims with mathematical elegance: the prose is clear and tight and the proofs are compact and pretty. To be sure, representation theory lends itself well to such a presentation (with Serre leading the way, of course), what with its mix of abstract and computational algebra, ersatz Fourier analytic methods (for the time being: nothing ersatz about what happens later — see e.g. Gel’fand, Graev, Piatetski-Shapiro, Representation Theory and Automorphic Functions), and (especially in the book under review) category theory; but there is no denying that the authors have done an excellent job in presenting the earlier parts of these themes to the reader.
Add to this the sets of problems included in the book, replete with occasional hints and estimates of the degree of difficulty, as well as the wonderful “Historical Interludes” by Slava Gerovitch (I particularly recommend the one about Hermann Weyl), and the result is a fantastic little book (a bit over 200 pages): I think it is bound to become the way to get into this subject “holistically” (as the book’s back cover has it).
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.