An alternating sign matrix is simple to describe. It is a square matrix whose only non-zero entries are +1s and –1s arranged so that the signs alternate within each row and each column, and such that the sum of the entries in each row and in each column is +1. Permutations are special cases of alternating sign matrices. Any permutation σ of n letters can be represented as such a matrix by setting aij = 1 if σ(i) = j and aij = 0 otherwise. The number of permutations of n letters is easily counted as n!, but what about the number of n × n alternating sign matrices? The subject of this book is the conjecture that there are
of these matrices.
In Proofs and Confirmations David Bressoud tells the story of the twenty year history of this conjecture. He does so with two goals in mind: to exposit the mathematics, and to share an example of the chronological development of research mathematics. As mathematicians know but students rarely recognize, proofs don't usually spring up perfectly formed. Here Bressoud introduces us to William Mills, David Robbins, and Howard Rumsey, who together made the original conjecture along with several refined conjectures. He follows their path into plane partitions and symmetric functions, picking up several more conjectures. There are a total of 14 conjectures presented (thankfully summarized for the reader early on in the book), most of which are proven here. The path eventually leads to the door of Doron Zeilberger, who proved the Alternating Sign Matrix conjecture in the mid 1990s. Not stopping there, Bressoud links alternating sign matrices with physicists' notion of "square ice" and proves a refined version of the conjecture.
As a mathematician whose expertise is outside the realm of combinatorics, I found this book to be enjoyable to read. (It proved to be a bit too much as bedtime reading, however, when all the formulae and summations tended to blur together.) Although there is quite a bit of terminology (e.g., plane partition, symmetric plane partition, symmetric self-complementary plane partition, totally symmetric self-complementary plane partition and so forth), Bressoud makes excellent use of examples to help the reader note the differences and clarify the concepts. In fact, I often found in reading that the author had anticipated my questions! Alas, the final chapter is too short; I would have been interested to learn more about square ice. Despite this wish, the final chapter did provide satisfaction: not only has the original conjecture been proven, but there is a connection to physics that I didn't expect to find.
While this isn't a book I will recommend to non-mathematician friends, it is a book for motivated undergraduate students to pursue as an independent study. I think this would also be appropriate as a text for a Topics of Mathematics class. The book jacket claims that it is accessible to anyone with a knowledge of linear algebra. That's true; however, I would suggest that the number and size of the equations will be construed as daunting to the average student. There are plenty of exercises — on average, thirteen — at the end of each section. While some of these exercises do invoke a computer algebra system (Mathematica), most do not. Many of the exercises point the reader to exactly those places in the section where pencil and paper are required if one hopes to gain an understanding of the material.
Bressoud has done a very nice job of presenting us with a readable book which delivers a self-contained look at some current mathematics. And he's done a wonderful job at exposing the flavor of research mathematics. Take a look.
Michele Intermont (email@example.com) is Assistant Professor of Mathematics at Kalamazoo College. Her area of specialty is algebraic topology.