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Symmetry, Representations, and Invariants

Roe Goodman and Nolan R. Wallach
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 255
[Reviewed by
Michael Berg
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I had a truly marvelous time in graduate school at UCSD in the early 1980s and still think back fondly to those days now long ago. There were so many things to do, to sample, to delve into, and there was so much faculty-student interaction to boot: we would find things out about everything under the mathematical sun just by hanging around, chatting over tea and cookies, and crashing seminars of apparent interest. To be sure, this scene is not unique to La Jolla — I guess much of it is the Princeton model transplanted to Southern California. (Is Princeton the originator of the afternoon teas? I think so…) But there was something very special in the air when I was at UCSD, possibly fostered by the uncommon amount of freedom allowed us, the graduate students, the relaxed if not quite laissez faire style of the faculty, and the flexibility of the department as such. Or maybe it was the fact that UCSD, abutting Torrey Pines Golf Course, a mile or two from La Jolla beaches, and equally proximate to gorgeous bluffs favored by a multitude of hang-gliders, felt more like a resort than anything else. To a large extent picking up mathematics in this environment was, as the song says, “like breathing out and breathing in.”

Of course this kind of cultural immersion is no substitute for the blood, sweat, and tears required to learn anything well, e.g., to a sufficient depth to facilitate the production of an acceptable doctoral dissertation: hard labor is called for, and hard labor we did, but the sting is certainly diminished by the luxury of being able to write the dissertation on the beach, as I did on many occasions.

And then, coming back to campus, you’d run the risk of spotting Michael Freedman (a very dedicated rock-climber) glued to the side of the Mathematics Building, several storeys up, or of walking in on a seminar series given by Alain Connes — at Gallic linguistic speed — aimed largely at Max Karoubi (and, yes, they might as well have been speaking French). And there was Yau and his entourage of students from both the local geometry talent pools and imported from China… All in all it was a microcosm of the mathematical world set in an idyllic locale.

But what does all this self-indulgent reminiscing have to do with Symmetry, Representations, and Invariants, by Roe Goodman and Nolan R. Wallach? The answer has two intertwined parts: the second author is a long-time member of the UCSD faculty, and the book evinces a style, and an ineffable quality, that fits hand-in-glove with the ethos of the UCSD Mathematics Department as I have tried to sketch it above. Deep things are presented with a light touch, complete and well-motivated, but without pedantry. And sometimes you do not know the great value of what you are getting till you reflect on it later.

Coming in at over 700 pages, Symmetry, Representations, and Invariants is really several books in one, again with intertwined parts. The authors suggest that Chapters 1–3 can serve in the cause of a one-semester course on Lie- and algebraic groups and classical representation theory; adding Chapter 11 to the mix makes for a thorough one-semester treatment of the characteristic 0 case. Then Chapters 3–4, 6–10 provide the material for a second course, or, alternatively, for the more geometrically inclined, the recipe would be Chapters 3–4, 11–12. Additionally, “[a] year-long course on representations and classical invariants along the lines of Weyl’s book would follow Chapters 1–5, 7–9, and 10.” A cornucopia, to be sure. (Happily Gooman and Wallach provide a full Leitfaden on p. xix.)

Furthermore, the authors are experienced veterans in the game of writing sound and useful books on the indicated subjects: witness their earlier Representations and Invariants of the Classical Groups. Say the authors: “[t]he parts of the previous book that have withstood the authors’ many revisions as they lectured from its material have been retained.” But Symmetry, Representations, and Invariants goes well beyond its predecessor, particularly as regards the book’s latter parts. To wit: “[t]he last two chapters… develop, via a basic introduction to complex algebraic groups, what has come to be called geometric invariant theory.”

I think this is a terrific book, succeeding in its considerable ambitions, and doing so in the remarkable style I tried to sketch above. Symmetry, Representations, and Invariants is indeed capable of instructing the reader in the three themes given in its title, taking him from elementary and foundational notions to very advanced material (consider e.g. § 5.6, which includes Howe duality; Ch. 8 on branching laws; § 10.3 on Riemannian curvature tensors; § 10.4 on knot polynomials and invariant theory; and § 12.4 and its coverage of work by B. Kostant and S. Rallis). There are over 350 exercises which the reader should certainly engage in combat, and there are extremely useful and ramified (if compact) appendices: algebraic geometry (sans sheaves — should we call this “pre-sheaf algebraic geometry” (with excuses to Grothendieck)?), the obligatory stuff on linear and multilinear algebra (which should be review material, of course), associative and Lie algebras, and finally manifolds and Lie groups (with Poincaré-Birkhoff-Witt appearing on p. 670).

Symmetry, Representations, and Invariants promises to be a graduate text of major importance.

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