An Introduction to the Representation Theory of Groups

Representation theory comes in many flavors, a lot like curry: it’s still curry, but there’s a notable difference between yellow and red curry (as part of the Thai school, as it were); and then there’s Indian curry…. The most natural, or arguably the most accessible, version is probably the representation theory of finite groups, or at least that’s how it starts out. Things get very sticky very quickly, with Brauer theory being a good illustration: it fits into the framework of the classification of finite simple groups, and we all know how sticky that subject is.

Then there is the representation theory of Lie groups, usually coupled with that of Lie algebras, famous as a subject of surpassing beauty and imposing scope. In this connection I might mention V. S. Varadarajan’s definitive book on this subject, i.e. his Lie Groups, Lie Algebras, and Their Representations. I am very fortunate to have been Varadarajan’s reading student as an undergraduate. This fabulous mathematician, truly effectively my undergraduate advisor (there were no such explicit designations at UCLA in the 1970s), has left a deep imprint on my entire mathematical life, in particular my areas of research. It was he who introduced me to the work of André Weil, which takes me to the third flavor of representation theory I want to mention, namely, unitary representation theory. It is of course closely allied to the aforementioned theme. In his famous 1964 Acta Math. paper, “Sur certains groups d’opérateurs unitaires,” Weil laid out the representation theory of the metaplectic group (a double cover of the symplectic group) and therewith presented a deep and exceedingly elegant (unitary) representation theoretic explication of C. L. Siegel’s analytic theory of quadratic forms. This landmark paper builds on the introduction of the oscillator representation in the context of quantum mechanics by David Shale following I. E. Segal, and does something that is still remarkably surprising to me: in this context Weil derived an equivalent to Hecke’s Fourier analytic proof of quadratic reciprocity by means of maneuvers that are part of what Weil would call abstract Fourier analysis.

Representation theory makes its impact felt in such ostensibly disparate areas like quantum physics and number theory and many besides (e.g., probability). Accordingly, it is pretty much non-negotiable that every mathematician should know something (substantial) about the subject in its different flavors, and this is what Kowalski is trying to foster in the book under review. Says he: “representation theory is a fundamental theory, both for its own sake and as a tool in many other fields of mathematics; the more one knows, understands, and breathes representation theory, the better.” His approach is refreshingly practical: “the objective has never been to give the shortest or slickest proof … the goal is rather to explain the ideas and the mechanism of thought that can lead to and understanding of ‘why’ something is true, and not simply to the quickest line-by-line check that it holds.” So, we’re dealing with some sound pedagogy and a treat for the (diligent) student.

Kowalski adds a minor disclaimer: “although I have tried to illustrate many aspects of representation theory, there remain topics that are barely mentioned or omitted altogether … [e.g., t]he representation theory of anything else than groups … in particular Lie algebras … [and] the precise classification of representations of compact Lie groups … [other than] SU₂(C) …” He provides references, though, including Fulton-Harris and Knapp.

All right, then, how does Kowalski go about his task? Well, briefly, the fundamental stuff is in Chapter 2, from the basic formalism to Clifford algebras; Chapter 4 is titled, auspiciously, “Linear representations of finite groups” (bringing to mind Serre’s classic by that title); and then it’s on to topological groups: Chapter 5 deals with compact groups and includes, for example, Peter-Weyl, and Chapter 6 elaborates on this important theme, starting off with the suggestive and critical section, “Compact Lie groups and matrix groups.” Unitary representations occur in Chapter 3, right after Kowalski’s official introduction to topological groups: I guess the middle of this chapter identifies something of a point of departure. The book has an appendix containing three sections: on algebraic integers, on the spectral theorem, and Stone-Weierstrass. Finally, SL₂(R) appears in Chapter 7, right after “locally compact abelian groups.” Wow. Quite a cornucopia.

It’s a good book: it’s very useful and well written. What with exercises scattered throughout, as well as examples and to-the-point remarks, it was clearly crafted with the student in mind. Kudos.

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