MAA Reviews

Group Theory: Birdtracks, Lie's, and Exceptional Groups

Predrag Cvitanović

Publisher: Princeton University Press (2008)
Details: 280 pages, Hardcover
Price: $39.95
ISBN: 9780691118369
Category: Monograph
Topics: Mathematical Physics, Lie Groups and Algebras

[Reviewed by Michael Berg, on 09/04/2008]

This book has to be seen to be believed! The title, Group Theory, is nothing if not surprising, given that the material dealt with by Predrag Cvitanović in these roughly 250 pages requires a level of sophistication well beyond what is offered in the early stages of university algebra. In point of fact, the general theme of the book under review is Lie theory with representation theory in the foreground, and Cvitanović’s revolutionary goal is to develop large parts of the subject strictly by means of calculi of diagrams (e.g., “birdtracks”) and, for lack of a better word, the attendant combinatorics.

Indeed, it is the book’s subtitle, Birdtracks, Lie’s, and Exceptional Groups, that gives at least a hint of what is to follow and that it will take us far off the beaten track: what is a birdtrack, after all? Well, we are quickly told that it is a “notation inspired by the Feynman diagrams of quantum field theory,” originally invented (in prototype) by Sir Roger Penrose. Birdtracks in fact present invariant tensors and “invariant tensor diagrams replace algebraic reasoning in carrying out all group theoretic computations… [The indicated] diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations.” It is for these reasons that Group Theory boasts more than 4000 diagrams and illustrations, thus yielding an average of sixteen per page. Zowie!

Given the preceding brief sketch of birdtracks’ raison d’être it is proper, then, to compare them, as suggested, to the most famous, and supremely successful, ploy for replacing gruesome if not prohibitive calculations by a yoga of diagrams, namely, Richard Feynman’s ultra-slick short-hand for doing perturbation computations (integration by parts on metabolic steroids) in Q(uantum) E(lectro) D(ynamics). Says Cvitanović on p. 42: “In developing the ‘birdtrack’ notation in 1975 I was inspired by Feynman diagrams and the elegance of Penrose’s ‘binors’… So why… ‘birdtracks’ and not ‘Feynman diagrams’? The difference is that here diagrams are not a mnemonic device, an aid in writing down an integral that is to be evaluated by other techniques… Here ‘birdtracks’ are everything — unlike Feynman diagrams, here all calculations are carried out in terms of birdtracks, from start to finish.”

This having been said, it is clear that the reader of this monograph should already be rather familiar with Lie groups and representation theory (I do like Serre’s old book for learning this gorgeous material) and be disposed to adopt an utterly pictorial way of doing calculations in this area. The fact that Cvitanović’s Group Theory is not intended for rookies is revealed right off the bat by the list of chapters. We find, on p. 5 (!), the following passage: “…the first seven chapters [of 21] are largely a compilation of definitions and general results that might appear unmotivated on first reading. The reader is advised to work through the examples… in [the second] chapter, jump to the topic of possible interest… and birdtrack if able or backtrack if necessary.” Obviously this sage advice is not aimed at a novice; it’s even fair to say, I think, that Cvitanović has the in-crowd of Lie theorists (or those aspiring accordingly) as his target audience.

Furthermore, this audience ought to be peppered with a decent number of physicists: consider, e.g., the following remarks on p. 166: “What are these ‘spinsters’? A trick for relating SO(n) antisymmetric reps to Sp(n) symmetric reps? That can be achieved without spinsters: indeed Penrose… had observed many years ago that SO(-2) yields Racah coefficients in a much more elegant manner than the usual angular momentum manipulations…” On the other hand, a few lines down the page we encounter metaplectic representations of the symplectic group, pointing to deep themes in analytic number theory (reciprocity laws for global number fields treated in the style of Weil and Kubota). It is worth mentioning in this connection that metaplectic covering groups , as such, originate in André Weil’s famous 1964 Acta Math. paper, “Sur certains groupes d’opérateurs unitaires,” which has the projective Weil representation at its core. But this paper was preceded by an article by I. E. Segal, introducing the prototype of this projective representation in a physics context, and nowadays it is often referred to as the Segal-Shale-Weil representation (when it’s not simply called the oscillator representation). In any event, this theme, as also the very orientation of Cvitanović’s Group Theory, properly belongs to the area where physics and mathematics meet.

Thus, for the right reader, which is to say, an R>0-linear combination of mathematician and physicist equipped with a zeal for novel combinatorics-flavored diagram-gymnastics, this book will be a treat and a thrill, and its new and radical way to compute many things Lie is bound to make its mark.

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