The Quantum MacMahon Master Theorem

By Stavros Garoufalidis, Thang TQ Le, Doron Zeilberger

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Appeared in Proc. Natl. Acad. of Sci. (PNAS) v. 103 (2006), No. 38 (Sept. 19), 13928-13931.

First Written: March 25, 2003.

Revised and Corrected Version: Oct. 8, 2004.

Added Oct. 8, 2004: The original version had an error. The mistake was that we tried to prove too little, and hence used too strong of an hypothesis in the statement of the theorem, for which a crucial lemma, about some matrix, was wrong, as was pointed out by the referee who showed that part of the stronger hypothesis did not hold for that matrix. Luckily, the lemma is nevertheles true for the weaker hypothesis, and surprise!, the theorem also holds with the strong hypthesis replaced by the weaker one. So all is well that ends well.
Download the Maple package qMM, that empirically proves the Quantum Macmahon Master Theorem for fixed r, and bounded degree, that empirically proves the Quantum Macmahon Master Theorem and the more general Maple package QuantumMACMAHON , that proves, empirically yet rigorously, the theorem for fixed dimension (in practice up to 5) but for any degree!
Added June 15, 2005: The version of qMM above has just been adapted from the previous version qMMfull, that assumed the full-quantum relation .
If you write a paper, and get excited about it, but no one else does, don't get discouraged. It may take 25 years for its significance to come out.

My second postdoctoral position was spent, in the AY 1978-1979, at Georgia Tech. During that year I wrote the article "The algebra of linear partial difference operators and its applications". It was inspired by I.J. Good's "proof from the book" of the Dyson conjecture, and put it in the context of difference operators, and tried to show its potential importance.

I sent copies to Andrews, Askey, Dyson, Good, and Riordan. While they all gave me polite responses, the only one who really liked it was I.J. Good. Also the Math Review by Dick Askey was on the lukewarm side. One of the criticisms, by George Andrews, was that the "q-analog" of MacMahon's Master Theorem, was indeed a `q-analog', but not the kind he meant, and it seems not to be useful for q-series.

George may have been right about the particular q-analog that I proposed there, but I was recently vindicated in my gut feeling that the paper was "important", since the method introduced there proved crucial for the present paper, that proves a beautiful and natural quantum analog, conjectured by Stavros Garoufalidis and Thang Le, that according to them has important potential applications in knot theory.

It is an amusing fact that Stavros is now at Georgia Tech, that became a much better department since my time.

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