By Stavros Garoufalidis, Thang TQ Le, Doron Zeilberger and X [maybe you!]
First Written: March 25, 2003.
I am offering a prize of $200 for fixing this gap, and the fixer will be a co-author, of course, provided that he or she will write the revised version. This way the author would get Garoufalidis , Le, and Zeilberger number 1's, and possibly decrease her or his Erdos , Einstein, and other numbers.
Here is an outline of what I believe should be the ultimate proof. The formula on line 11 from the bottom of p. 3 must be correct, since it was checked by computer up to r=6. Since it implies the Quantum MacMahon theorem, we do have a proof for dimension up to 6, and with bigger computers one can prove it for larger r, but not for all r. For all r, we still need an additional human idea.
You can expand the determinant (that is by itself not quantum, but use the definition of quantum-determinant), and a get a linear combination of terms of the form a[i_1,j_1]a[i_2,_j2]...a[i_s,j_s] times [M_{k_1} M_{k_2}... M_{k_{r-s)}H]. Here {i_1, ..., i_s} is any subset of {1,..., r}, [j_1,..., j_s] is a permutation of [i_1, ..., i_s] and k_1>k_2>...>k_{r-s}, where {k_1, ..., k_{r-s}} is the complement of {i_1, ..., i_s} w.r.t. {1, ..., r}. Now M_{k_1} M_{k_2}... M_{k_{r-s)}H= Some Quantum polynmial in the a[i,j]'s and (Laurent in the x[i]'s) times H. (Use the fact that the operators P_i of the middle of p.3 annihilates H(m_1, ..., m_r; x_1, ..., x_r), to rewrite it as M_iH= Expression(a[i,j]'s, x[i]'s, m[i]'s)H, and then iterate. So for any specific r, the recurrence
det_q (M_i delta_{i,j} - a_{i,j})H =0
is just a routine quantum-high-school-algebra verification, that has been performed by Shalosh for r=1,2,3,4,5,6 by typing ProveqMM(r); in the Maple package QuantumMACMAHON .
Some suggestions: 1) Quantum-linear-algebra approach: Use deeper properties of quantum-linear algebra, and possibly an appropriate change-of-variables, to convert this expression to a quantum-determinant of a (quantum!) matrix of lower-than-r rank (whatever it means). This would be along the lines of my 1978 proof (ref. [Z]).
2) Each term has an obvious combinatorial interpretation as the weight-enumerator of a partial-permutation and a graph with every vertex not in the partial permutation having out-degree 1, with the obvious weight. However, in this case, one has to also use the quantum-commutation rules, and each such creature gives rise to many more new creatures, if you want to bring every monomial to canonical form. Study these ultimate combinatorial creatures and find a sign-reversing involution.
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.