Introduction to Group Characters

When I was in graduate school starting my doctoral thesis in number theory it soon became obvious that I needed a crash course in representation theory, which is to say that I needed to find a few good sources from which to learn what I needed to know for my thesis problem as expeditiously as possible. I found two books to be of particular use, both of them coming from “across the Pond,” as the saying goes. One was no. 34 in the LMS Lecture note series, titled, Representation Theory of Lie Groups, a series of workshop articles (as we’d now call them) edited by Michael Atiyah, and the other was the first edition of the book under review, Ledermann’s Introduction to Group Characters. I have fond memories of both of these books, and it is a pleasure to get a chance to look at Ledermann’s book again.

The present second edition, appearing a couple of years after I finished my doctorate, testifies by its very existence to the success of Ledermann’s enterprise. The author notes that his newer edition includes “further work on tensor products, arithmetical properties of character values and the criterion for real representations due to Frobenius and Schur.” This material is added to his existing menu of excellent choices, making for a very good introduction to representation theory. I should like to counsel, however, that the novice in this subject couple his studies of this material to, say, some Lie group representation theory: Ledermann restricts his focus to “characters of a finite group over the complex field.”

Furthermore, Ledermann’s focus on character theory might also be supplemented by a perusal (and then some) of, say, Serre’s Linear Representations of Finite Groups, given that in this treatment the focus falls on what one might call the homological algebraic aspects of representation theory. These complementary perspectives, Ledermann’s and Serre’s, should make for a wonderful way to prepare for more advanced or broader themes in representation theory of finite groups. And, again, the reader should also not neglect the infinite groups, or specifically the usual Lie groups and the theory of unitary representations in a Hilbert space. But that material comes later than, or at least separately from, what is given by Ledermann.

Ledermann’s book is also distinguished by an interesting and evocative historical connection. Says the author: “My own interest in the subject goes back to an inspiring course by Issai Schur which I attended in 1931 … Occasionally, Schur would enliven lectures with anecdotes about his illustrious teacher Frobenius, and I may be forgiven if I have succumbed to a bias in favour of an ancestral tradition.” (I should prefer to applaud such a bias, actually…)

Ledermann’s Introduction to Group Characters is extremely accessible and a pleasure to read (even without my own sentimental predisposition). The presentation is extremely clear and is very comfortably paced: induced characters don’t appear till p.69, with Frobenius reciprocity occurring on p.74, and Mackey’s theorem is found on p.94 (preceded by its derivation, its statement is on p.97). Algebraic number theory makes its appearance in the section titled “Group-theoretical applications,” which also includes Burnside’s (p,q)-theorem, and the book’s final chapter, “Real representation,” is formed around a wonderful leading question: given an irreducible representation F of a finite group, whose coefficients are à priori allowed to be complex numbers (for some group elements), does there exist a non-singular matrix T such that the intertwining T^-1FT is always real, regardless of what group element it is evaluated on?

Well, four pages after Ledermann poses the question he gives us the beautiful theorem of Frobenius and Schur, referred to earlier as their “criterion for real representations,” to the effect that this is always the case if the representation is orthogonal, i.e. for each group element the representing matrix is orthogonal.

Introduction to Group Characters is also peppered with a decent quota of good exercises, and, as always, the diligent student should go at them with zeal and commitment. It’s quite a good book!

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