Bressoud D.M. Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture

Cambridge: Cambridge University Press, 1999. — 290 p.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 n × 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.
The book is accessible to anyone with a knowledge of linear algebra. Students will learn what mathematicians actually do, and even researchers in combinatorics will find something new here.The ConjectureHow many are there?
Connections to plane partitions
Descending plane partitionsFundamental StructuresGenerating functions
Partitions
Recursive formulæ
DeterminantsLattice Paths and Plane PartitionsLattice paths
Inversion numbers
Plane partitions
Cyclically symmetric plane partitions
Dodgson’s algorithmSymmetric FunctionsSchur functions
Semistandard tableaux
Proof of the MacMahon conjectureHypergeometric SeriesMills, Robbins, and Rumsey’s bright idea
Identities for hypergeometric series
Proof of the Macdonald conjectureExplorationsCharting the territory
Totally symmetric self-complementary plane partitions
Proof of the ASM conjectureSquare IceInsights from statistical mechanics
Baxter’s triangle-to-triangle relation
Proof of the refined ASM conjecture
ForwardGeneral Index