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

ISBN-13: 9780521666466 / Angielski / Miękka / 1999 / 292 str.

David M. Bressoud; William Watkins; Gerald L. Alexanderson

Opis

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.

Recenzje

'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

Spis treści

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