88b:11063
Andrews, George E.
$q$-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra.
CBMS Regional Conference Series in Mathematics, 66.
Published for the Conference Board of the Mathematical Sciences, Washington, D.C.; by the American Mathematical Society, Providence, R.I., 1986. xii+130 pp. $16.00. ISBN 0-8218-0716-1

The theory of partitions [Addison-Wesley, Reading, Mass., 1976; MR 58 #27738] by the author has been one of the most widely read and used books in the series "Encyclopedia of Mathematics and Its Applications". It appeared on the eve of a tremendous burst of activity in partition theory and has become the standard introduction for people who would like to enter this field. The chief complaint that might now be levelled against it is that it was published too long ago and makes no mention of many of the techniques and problems that are around today.

The book under review could well have been entitled "The theory of partitions, Part II". It is not and does not claim to be an exhaustive compendium of everything that has happened in partition theory in the past decade. But it starts where The theory of partitions ends and develops several lines of research with which the author has been involved. For the mathematician having a nodding acquaintance with partition theory or one merely intrigued by it, this book will serve as an excellent introduction to the beauty, diversity and power of what has been accomplished since 1976.

As the title suggests, one of the revolutions in partition theory has been the range of its applicability. This is reflected in discussions of Baxter's hard hexagon model in statistical mechanics, the Macdonald conjectures and Selberg's integral generalizing beta-integral evaluations, the Lusztig-Macdonald-Wall conjectures, special functions and Hecke modular form identities. This book also includes an introduction to Ramanujan's work in the "Lost Notebook", the forgotten contributions of L. J. Rogers, and descriptions of some of the new tools of the trade, including the Garsia-Milne involution principle and the Bailey chain, among many other topics touched on briefly.

One of the most striking features of this book is the pervasiveness of computer algebra. The final chapter is the magician revealing his tricks as Andrews shows how the incredible formulae that crop up in this book were or could have been conjectured by using today's symbolic manipulation packages. The argument is very powerfully presented through example that the computer has become an essential tool of the research mathematician.

This book is extremely readable. The chapters stand on their own and can be dipped into by anyone interested in just one problem or technique. Still, much is left out. There is no mention of the very rich ties between partition theory and representation theory, nor of recent connections with cohomology theory. Those are stories for someone else to write.

Reviewed by David M. Bressoud