Proofs and Confirmations
The Story of the Alternating-Sign Matrix Conjecture

An introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses.

David M. Bressoud (Author)

9780521666466, Cambridge University Press

Paperback, published 13 August 1999

292 pages, 49 b/w illus. 307 exercises
22.9 x 15.2 x 1.7 cm, 0.43 kg

'This is an excellent book which can be recommended without hesitation, not only to specialists in the field, but to any mathematician with time to read something interesting and nicely written.' EMS

This is an introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of invariant theory, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1996 proof of the original conjecture. The book is accessible to anyone with a knowledge of linear algebra. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something new here.

1. The conjecture
2. Fundamental structures
3. Lattice paths and plane partitions
4. Symmetric functions
5. Hypergeometric series
6. Explorations
7. Square ice.

Subject Areas: Mathematical logic [PBCD]