Proof of the refined alternating sign matrix conjecture

(Appeared in `` New York J. of Mathematics'' (v. 2 (1996), 59-68.)

Even though the first version of my paper ``Proof of the alternating sign matrix conjecture'' was written at the end of 1992, it took more than two years, and more than eighty checkers, to make it formally correct and watertight. Consequently, it was only formally accepted (by the Elect. J. of Comb.) May 1995. Because the final version was so long (80 pages), I was sure that it would be a long time before another proof would be found. I was very surprised when, at Aug. 29, 1995, Greg Kuperberg observered that it is an almost immediate consequence (modulo routine computations) of the Izergin-Korepin formula, that, in turn, follows from the Yang-Baxter relations. He wrote this up in: ``Another proof of the alternating sign matrix conjecture''. (Appeared in Inter. Math. Res. Notes 1996).

I quickly realized that the same method of proof could be used to prove a refined, stronger, version of the ASM conj., also conjectured by Mills, Robbins, and Rumsey. I like this proof very much since it is the first time that I got to actually use the orthogonality of ``classical orthogonal polynomials''.

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Added June 8, 1999: Dave Bressoud and Jim Propp wrote a delightful and fascinating account about the proofs of the ASM conjecture and their ramification.

It appeared in The June/July 1999 issue of the Notices of the American Mathematical Society.

Dave Bressoud's beautiful book about this will appear this summer.

Added Feb. 8, 2000: Dave Bressoud's beautiful book, `Proofs and Confirmation: The Story of the Alternating Sign Matrix Conjecture", published jointly by Cambridge and MAA, came out at the end of Summer 1999, and has very deservedly won the MAA's Beckendorf book award.

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