A Tour of Representation Theory

This excellent book is one that I wish had been written when I was a student. My exposure to representation theory was gradual and segmented, extending over a period of years. I first learned about the subject in an undergraduate special topics course on group representations that was very down-to-earth and matrix-based. Later, as a graduate student, I encountered the subject again in my first-year algebra course, from a more sophisticated perspective: Maschke’s theorem, for example, was no longer presented as a result about block matrices but instead was formulated in terms of the semisimplicity of a certain ring. Even then, however, I still associated the term “representation theory” exclusively with finite groups. Then, still later, when I began to get interested in Lie algebras as a dissertation topic, I read Humphrey’s book (Introduction to Lie Algebras and Representation Theory; now a classic, but quite new back then) and learned that there is a representation theory of these objects as well. Then, after leaving graduate school, I learned about other objects that can be the study of representation theory, including Hopf algebras and quivers.

Had I the benefit of a book like this one (hereafter denoted Tour) in my early graduate years, I could have saved myself a lot of time. This book, true to its title, surveys the field of representation theory, offering valuable looks at the distinct flavors of the subject, though also stressing its unifying features and interconnections. Its intended audience is second-year graduate students (a year of graduate algebra, including a very good background in linear algebra, is the official prerequisite), but the book should also be of considerable interest to faculty members who want a look at what’s going on in this area.

This is not the only book that offers a broad survey of representation theory; there is also, for example, Introduction to Representation Theory by Etingof et al. The book now under review, however, is substantially different from Etingof’s book. It is aimed at a more sophisticated audience, is about three or four times as big, is far more comprehensive and detailed, and has somewhat different topic coverage: the Etingof book discusses groups, Lie algebras and quivers; Lorenz’s book discusses these first two topics but covers Hopf algebras rather than quivers. It is also much better suited for use as a text, because, if nothing else, it contains many more exercises (about which, more later) than does the Etingof book.

Another, quite recent, addition to the survey-of-represention-theory field is A Journey Through Representation Theory by Gruson and Serganova. (So we have both a Tour and a Journey; can a Cruise Through Representation Theory be far behind?) Here again, although this book has some similarities with the book now under review, Lorenz’s book is quite different: it is much larger than Journey, which is only about 200 pages long, and also has different topic coverage: Journey (unlike Tour) discusses quivers and compact groups as well as finite group representation theory; the representation theory of Lie algebras, however, is only mentioned briefly at the end of the text. (Lorenz does not discuss compact groups in this text because he wants to keep it, in his words, “resolutely algebraic”.)

Tour is a large book, weighing in at more than 650 pages, and there is somewhat more material in it than can be covered in a two-semester class, but there is considerable independence among its four separately-numbered parts, thus offering a prospective instructor quite a lot of flexibility in its use.   The first part looks at algebras and their representations. There are two chapters in this part: the first discusses algebras that will be important in the sequel, and introduces the basic notions of representation theory, such as semisimplicity and the Weddeburn structure theorem. The second chapter covers projective modules and Frobenius algebras and is not quite as essential to what follows.

Part II of the book is on the representation theory of groups, mostly but not always finite. The material of part I of the text enters the discussion here because a representation of a finite group \( G \) over a field \( k \) can be identified with a representation of the group algebra \( kG \).  Thus, for example, as noted above, Maschke’s theorem, in this context, is just the statement that \( kG \) is a semisimple algebra if the characteristic of \( k \) does not divide the order of \( G \)(which of course is always true if this characteristic is 0).

Although Part II of the text is only two chapters and roughly 130 pages long, the amount of material presented is substantial. The first chapter is on general group representation theory. It covers all the “usual suspects” in an introductory look at group representation theory: irreducible representations, character tables, Maschke’s theorem, etc. Induced representations are developed in an exercise, based on the induction functor introduced in Part I of the book. Several purely group-theoretic results (including Burnside’s  \(p^{\alpha}q^{\beta} \) theorem) with representation-theoretic proofs are given. The notion of a Hopf algebra is briefly alluded to, in anticipation of the more thorough treatment given later, in part IV of the text.

Chapter 4, the second chapter in Part II, is on the representation theory of the symmetric group. The presentation in this chapter is somewhat novel and, the author tells us, is based on work of Okounkov and Vershik that dates back to a paper of theirs published in 1996. This material is not, I think, common in the textbook literature at this level. One book that does cover this approach is Linear and Projective Representations of the Symmetric Group by Kleshchev; Lorenz tells us that his presentation is based on that of chapter 2 in this book, but with more details and background material supplied.

Part III of the book, the longest part of the text, is devoted to the representation theory of Lie algebras. Because no prior familiarity with Lie algebras is assumed, the author starts from scratch with the definition, basic examples and elementary properties of these objects. Quite a lot of the general theory of Lie algebras is included here: for example, the classification of simple Lie algebras over an algebraically closed field of characteristic 0, using root systems and Dynkin diagrams, is discussed, though not completely proved; not unreasonably, the lengthy details of the classification theorem of connected Dynkin diagrams are omitted.  

In view of the depth of the discussion of Lie algebras, and the fact that the author has tried to make Part III fairly independent of Part II, this part of the book could, I think, serve as a text for a one-semester introductory graduate course in Lie algebras. It would not be an exaggeration to say that most of the topics covered in the first four chapters of Humphreys (which are basically devoted to introducing Lie algebras and discussing the classification theory of semisimple ones) also appear here; those four chapters probably amount to almost a semester’s work. Of course, the focus of the Lorenz text is on the representation theory of Lie algebras, and as a result the universal enveloping algebra (which plays roughly the role for this theory that the group algebra does for the representation theory of finite groups) is introduced somewhat earlier than it is, say, in the book by Humphreys.

Unfortunately, this part of Tour is marred by a serious printing error. (At least, this was the case in my copy of the book; I don’t know how widespread the problem is.) Page 310 is followed by page 327; we then proceed from page 327 to page 334, which is then followed by page 327 again. From this new beginning, the pages seem to run normally. Thus, pages 311 through 326 are nowhere to be found.

Finally, part IV of the text is on representations of Hopf algebras, and also includes a chapter on linear algebraic groups. This is probably the most demanding and technical part of the book, and, as the author points out, is also the part that brings the reader the closest to the frontiers of current research.

There are also five appendices, all of which rapidly survey background areas in mathematics. Topics covered include category theory (the basic language of which is used in the text, but not to the point where it becomes overwhelming), linear algebra (mostly tensor products and the Hom functor), and commutative algebra (including elementary topics in algebraic geometry such as the Nullstellensatz and the Zariski topology). Other appendices discuss the diamond lemma (using it to prove the PBW theorem of Lie algebras) and the symmetric ring of quotients.

Finally, there is an extensive bibliography—ten pages of small type, containing 218 entries. The entries include both textbooks and journal articles, not all in English.

This is a very nicely written book, with student motivation always in mind. The level of difficulty increases as the book proceeds (as is only reasonable) but at no point does the book become too difficult for a well-prepared graduate student reader. There are lots of examples, as well as many exercises, of varying levels of difficulty. Solutions are not provided in the text itself (a pedagogical plus, in my opinion) but the author has prepared a solutions manual that he will make available to instructors who request it.  The copy of the manual that he was nice enough to supply to me is itself a substantial piece of work: it is 137 pages long and contains solutions (sometimes just hints, sometimes more detailed solutions) to many but not all of the problems in the book. In the manual, the author states that he hopes to gradually add to it, so the current manual may even be bigger and more detailed than the August 2018 version that was supplied to me.

I like this book a lot, and consider it to be a very valuable addition to the existing textbook literature on representation theory. It would not surprise me if it becomes the market leader in books on graduate-level representation theory.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.