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

This introduction to recent developments in algebraic combinatorics illustrates how research in mathematics actually progresses. The author recounts the dramatic search for and discovery of a proof of a counting formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While it was apparent that the conjecture must be true, the proof was elusive. As a result, researchers became drawn to this problem and made connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young; to partitions and plane partitions; to symmetric functions; to hypergeometric and basic hypergeometric series; and, finally, to the six-vertex model of statistical mechanics. This volume is accessible to anyone with a knowledge of linear algebra, and it includes extensive exercises and Mathematica programs to help facilitate personal exploration. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something unique within Proofs and Confirmations. 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. David M. Bressoud. Includes Bibliographical References (p. 261-268) And Index. Cover Title page Preface 1 The Conjecture 1.1 How many are there? 1.2 Connections to plane partitions 1.3 Descending plane partitions 2 Fundamental Structures 2.1 Generating functions 2.2 Partitions 2.3 Recursive formulre 2.4 Determinants 3 Lattice Paths and Plane Partitions 3.1 Lattice paths 3.2 Inversion numbers 3.3 Plane partitions 3.4 Cyclically symmetric plane partitions 3.5 Dodgson's algorithm 4 Symmetric Functions 4.1 Schur functions 4.2 Semistandard tableaux 4.3 Proof of the MacMahon conjecture 5 Hypergeometric Series 5.1 Mills, Robbins, and Rumsey's bright idea 5.2 Identities for hypergeometric series 5.3 Proof of the Macdonald conjecture 6 Explorations 6.1 Charting the territory 6.2 Totally symmetric self-complementary plane partitions 6.3 Proof of the ASM conjecture 7 Square Ice 7.1 Insights from statistical mechanics 7.2 Baxter's triangle-to-triangle relation 7.3 Proof of the refined ASM conjecture 7.4 Forward Bibliography Index of Notation General Index Photos (mediocre) "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 early 1980s: the number of m x n alternating sign matrices, objects that generalize permutation matrices. Although it was soon apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young, to partitions and plane partitions, 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 1995 proof of the original conjecture."--BOOK JACKET. "The book is accessible to anyone with a knowledge of linear algebra."--BOOK JACKET. This introduction to recent developments in algebraic combinatorics also illustrates how research in mathematics actually progresses.