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

by David M. Bressoud

• ISBN13: 9780521666466
• ISBN10: 0521666465
• Format: Paperback
• Copyright: 1999-08-13
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	xi
The Conjecture	1	(32)
How many are there?	3	(6)
Connections to plane partitions	9	(9)
Descending plane partitions	18	(15)
Fundamental Structures	33	(40)
Generating functions	35	(7)
Partitions	42	(13)
Recursive formulae	55	(8)
Determinants	63	(10)
Lattice Paths and Plane Partitions	73	(46)
Lattice paths	74	(10)
Inversion numbers	84	(8)
Plane partitions	92	(9)
Cyclically symmetric plane partitions	101	(10)
Dodgson's algorithm	111	(8)
Symmetric Functions	119	(32)
Schur functions	120	(7)
Semistandard tableaux	127	(7)
Proof of the MacMahon conjecture	134	(17)
Hypergeometric Series	151	(40)
Mills, Robbins, and Rumsey's bright idea	151	(9)
Identities for hypergeometric series	160	(17)
Proof of the Macdonald conjecture	177	(14)
Explorations	191	(32)
Charting the territory	191	(12)
Totally symmetric self-complementary plane partitions	203	(12)
Proof of the ASM conjecture	215	(8)
Square Ice	223	(38)
Insights from statistical mechanics	224	(8)
Baxter's triangle-to-triangle relation	232	(13)
Proof of the refined ASM conjecture	245	(12)
Forward	257	(4)
Bibliography	261	(8)
Index of Notation	269	(2)
General Index	271