Basic group representation theory is a subject that can, and probably should, be taught to junior or senior mathematics majors. But it rarely is. At its most elementary level, the subject involves homomorphisms from groups into groups of nonsingular matrices (or, what amounts to the same thing, into groups of invertible linear transformations from a finite-dimensional vector space V to itself), and therefore should, in theory, be accessible to any student with a reasonably good background in group theory and linear algebra. True, the subject then evolves into more sophisticated concepts (ring theory, tensor products, algebras, quivers, etc.) and, perhaps because group representation theory ultimately winds up at these destinations, most textbooks on the subject make extensive use of them and are, for this or other reasons, not really suitable for undergraduates.
A quick (and non-exhaustive) survey of the existing literature should illustrate this point. Serre’s Linear Representations of Finite Groups is concisely and elegantly written but beyond the reach of most undergraduates; even the first (and most accessible) third of the book mentions such topics as compact groups and invariant measures. Weintraub’s Representation Theory of Finite Groups: Algebra and Arithmetic is also intended for a graduate audience (it appear in the AMS Graduate Studies in Mathematics series) and, as explained in the preface, a goal of the book is to discuss representation theory in a fairly general context; the very first substantive chapter, after a six-page introduction, is entitled “Semisimple Rings and Modules”. Likewise, the recent book by Sengupta, Representing Finite Groups: A Semisimple Introduction, is intended for graduate students and (not surprisingly, given the name of the book) makes use of semisimple rings and their modules.
Even a book that appears to be intended as a text for an undergraduate (well, British undergraduate) audience, James and Liebeck’s Representations and Characters of Groups, makes extensive use of modules and also uses tensor products; similarly, the book by Etingof et al, Introduction to Representation Theory, which has been used as an undergraduate text at MIT, has as a stated goal to give a “holistic” introduction to the subject which treats the representation theory of finite groups, Lie algebras and quivers as special cases of the general theory of representations of associative algebras.
The only book I am familiar with that is truly elementary is Victor Hill’s Groups and Characters, which does not even assume any prior background in group theory and whose representation theory discussions are firmly grounded in matrix groups. Unfortunately, Hill achieves simplicity in many of his discussions by simply omitting proofs of most of the more complicated theorems, and emphasizing examples throughout; as a result, the book, though informative, may be considered by some as unsuitable as a text for an upper-division course in representation theory.
In view of this quick survey, it should be apparent that the book by Steinberg now under review, which is based on a course offered at Carleton University that was populated mostly by senior undergraduates and first-year graduate students, and which avoids any sophisticated algebraic topics such as semisimple ring theory and tensor products, clearly fills an existing gap in the available textbook literature. It is also a pleasure to read — the exposition is concise (only about 150 pages of textual material) but quite clear; the author has obviously taken great pains to make the book accessible to undergraduates, even including a chapter-length review of the necessary linear algebra (inner product spaces and their operators, minimal and characteristic polynomials, diagonalization, the Spectral theorem, etc.) that the student may not have previously been exposed to.
Following this review of linear algebra, two chapters proceed through the definition and basic properties of group representations and characters, including such familiar topics as Schur’s Lemma, Maschke’s theorem, the formula for the sum of the squares of the dimensions of the irreducible representations, the orthogonality relations, and character tables.
All this is done under a standing assumption that the underlying field is the field of complex numbers, thereby avoiding complications with fields of prime characteristic or fields that are not algebraically closed. This is a reasonable assumption to make for a book at this level, but there is always some risk involved in making such a standing assumption and then not articulating the hypotheses in subsequent theorems; a “drop-in” reader of this book, for example, who looks up the statement of Maschke’s theorem on page 23 (“[e]very representation of a finite group is completely reducible”) without being aware of the standing assumption, would not realize that this theorem fails in some cases when the underlying field has prime characteristic. Likewise, the author’s statement of Schur’s Lemma does not mention that everything takes place over the field of complex numbers, but in the proof, when the algebraic closure of that field is used, the author does explicitly state “here is where we use that we are working over C and not R.”
Applications of representation theory are not slighted. Chapter 5 is on Fourier analysis on finite groups (including an application to graph theory), which is then applied in the final chapter of the book (chapter 11) to a discussion of probability and random walks in groups. This chapter, the contents of which were new to me, struck me as one of the more unusual features of this book.
Chapter 5 is followed by a chapter on an algebraic application of representation theory: a proof of Burnside’s Theorem (that no nonabelian group of order paqb, where p and q are primes, is simple). This famous result in group theory, which the author calls the “first major triumph of representation theory”, though not itself a statement about representations or characters, can be given a relatively accessible proof using group representations (along with some facts about algebraic integers that the author develops from scratch); it can also be proved without representation theory, but at the cost of considerable added difficulty (as well as passage of time: as the author points out, a proof avoiding representation theory took almost 70 years to discover). Later on in the book, another theorem of Burnside about the number of conjugacy classes in groups of odd order (a theorem that on its face has nothing to do with representations) is also proved using representation-theoretic methods.
Subsequent chapters extend the theory. Among the topics covered are permutation representations (preceded by a discussion of group actions), induced representations (wherein one “lifts” a representation of a subgroup H of G to a representation of G) and Frobenius reciprocity, Mackey’s criterion for irreducibility of an induced representation, real characters (and that other theorem of Burnside mentioned in the preceding paragraph), and an introduction to representations of the symmetric group using Young diagrams and tableaux.
These topics make for a very solid introductory course in group representation theory, particularly at the undergraduate level. Of course, to keep a book reasonably short, some selections have to made, and one might not agree with all of them. I would, for example, have preferred to have seen a bit more done with character tables. There are only about half a dozen that are worked out in the text, for example, and the reader is not given as much of an indication as I would have liked of how information about the group can be read off the table. In addition, the standard example of non-isomorphic groups with the same character table (the dihedral group D4 and the quaternion group) does not seem to be explicitly pointed out, although the character table for the quaternion group does appear (first as an exercise on page 50 and then worked out in the text on pages 102–103 using induced representations), and construction of the character table of D4 appears as an exercise on page 108.
By the way, neither the phrase “quaternion group” nor “dihedral group” appears in the Index. It also omits “Maschke’s theorem”, “Schur’s Lemma”, and both of the group-theoretic results of Burnside mentioned earlier. In other words, the index could be beefed up a bit.
These minor quibbles aside, the author has, by combining clear writing with an accessible and minimal-prerequisite approach to group representations, created a book that may well help bring group representation theory into the undergraduate curriculum. This is an impressive and useful text, and should be looked at by anybody with an interest in the subject.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.