MAA Blog: The q-Dyson Theorem

By: David Bressoud @dbressoud

David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences

I spent the fall of 1983 at the University of Minnesota supported by a Sloan Fellowship. Two important things happened that fall. The first was that I met Jan Alford, a returning adult student, at the Episcopal Center on campus. As Jan likes to say, God introduced us. A year later she joined me in Pennsylvania, and a year after that we were married. The other was one of my proudest contributions to mathematics, working with Doron Zeilberger on the proof of the q-extension of Dyson’s constant term identity, a result that George Andrews had conjectured in “Problems and Prospects for Basic Hypergeometric Functions,” one of the articles toward which Dick Askey had pointed me in the summer of 1976.

The collaboration with Zeilberger was slightly unusual. That summer Zeilberger circulated his draft of a complete proof of the Andrews conjecture. Doron is brilliant, enthusiastic, and impulsive. I have long admired his ability to piece together long, complicated arguments. That was certainly the case with this draft. Fortunately, I had the time and the interest that fall to slowly and methodically go through his argument. I found a lacuna, an assertion that he made that was not quite correct. How to correct it so that the proof could proceed was not at all obvious. I spent some weeks on that point, eventually saw how to fill in that gap, and then contacted Doron. We agreed that this would be a joint publication. It is traditional in mathematics that names appear in alphabetical order. But in this case the structure of the proof and almost all of the ideas were Doron’s. To emphasize this, we agreed that his name would come first. Our proof appeared in 1985 in Discrete Mathematics.

As Ian Goulden quickly realized, the core of this proof established a result of wide applicability. Doron was done with this problem and had moved on, so Ian and I explored some of these applications. One of them appeared in the Transactions of the American Mathematical Society and another in Communications in Mathematical Physics: “The generalized plasma in one dimension,” a result in statistical mechanics. The latter was an interesting experience, my only publication in a physics journal. In physics you publish short—they cut out all of my careful explanation of how we arrived at our conclusions—and you publish fast. Turnaround time was just a few weeks.

I will give a brief explanation of the q-Dyson Theorem after explaining Freeman Dyson’s original conjecture and presenting two very clever proofs.

Problem: Prove Dyson’s constant term identity (see Figure 1).

George Andrews (born 1938) and Freeman Dyson (1923–2020)

Dyson posed this conjecture in 1962 in the Journal of Mathematical Physics. It is a generalization of a result that he had needed and proven for his work in statistical mechanics. Kenneth Wilson supplied a proof later that same year, and still later in 1962 J. Gunson stated that he had also, and independently, found Wilson’s proof. Any constant term identity can be cast as the evaluation of a definite integral, the approach used by both Wilson and Gunson, but Dyson’s identity cries out for a simpler explanation. This was provided by the mathematician, cryptologist, and statistician I.J. Good in 1970, what is now known as “The Good Proof.” It is based on the Lagrange interpolation formula and explained in Figure 2.

Ira Gessel (born 1951) and Doron Zeilberger (born 1950).

Andrews’ conjecture introduces an extra parameter, q, into this constant term identity (Figure 3).

The multinomial coefficient arises naturally in combinatorics: (a_1 + a_2 + … + a_n)!/(a_1! a_2! … a_n!) counts the number of words using a_1 1’s, a_2 2’s, …, a_n n’s. A natural question is whether Dyson’s identity can be proven purely combinatorial. Doron Zeilberger accomplished this in 1981, drawing for inspiration from a combinatorial proof of the Vandermonde Determinant Formula that Ira Gessel had published in 1979 (Figures 3 and 4). Doron’s proof of Dyson’s identity is given in Figures 5, 6, and 7

Figures 3 and 4. Gessel’s proof of the Vandermonde determinant formula.

Figures 5–7. Zeilberger’s proof of the Dyson conjecture.

George Andrews’ conjecture is a q-analog of Dyson’s constant term identity, given in Figure 8. The proof that Doron and I constructed was based on his proof of the Dyson conjecture, only now each competition between two competitors carries a weight, expressed as a power of q. The weight of a particular multi-tournament is the product of the weights of all of the individual competitions. In 2006, Ira Gessel and Xin Guoce (family name first) produced an elegant proof based on the idea behind Cauchy’s proof of the Vandermonde determinant formula, that two polynomials must be be same if they are both of degree at most d and agree at d+1 points. In this case they treat each side as a polynomial in q^(a_0). The idea is simple. The details show great ingenuity. An even simpler proof was published in 2014 by Gyula Károlyi and Zoltan Lóránt Nagy that draws its inspiration from I.J. Good’s proof of the Dyson identity.

Figure 8. Andrews’ Constant Term Identity

References

Andrews, G.E. (1975). Problems and prospects for basic hypergeometric functions, pp. 191–224 in Theory and Application of Special Functions (R. A. Askey, ed.), Academic Press, New York,

Bressoud, D. M.; Goulden, I. P. (1985) Constant term identities extending the q-Dyson theorem. Trans. Amer. Math. Soc. 291 no. 1, 203–228.

Bressoud, D. M.; Goulden, I. P. (1987) The generalized plasma in one dimension: evaluation of a partition function. Comm. Math. Phys. 110, no. 2, 287–291.

F. J. Dyson, F.J., (1962). Statistical theory of the energy levels of complex systems. I, J. Math. Physics 3, 140–156.

Gessel, I. (1979). Tournaments and Vandermonde's determinant. J. Graph Theory 3, no. 3, 305–307

Gessel, I.M. and Xin, G. (2006). A short proof of the Zeilberger-Bressoud q-Dyson theorem. Proc. Amer. Math. Soc. 134, no. 8, 2179–2187.

Good, I. J. (1970). Short proof of a conjecture of Dyson, J. Math. Phys. 11, 1884

Gunson, J. (1962). Proof of a conjecture by Dyson in the statistical theory of energy levels, J. Math. Phys. 3, 752–753.

Károlyi, G. and Nagy, Z. L., (2014). A simple proof of the Zeilberger-Bressoud q-Dyson theorem, Proc. AMS. 142 (9), 3007–3011,

Wilson, K.G. (1962). Proof of a conjecture by Dyson, J. Math. Phys. 3, 1040–1043.

Zeilberger, D. (1982) A combinatorial proof of Dyson's conjecture, Discrete Math. 41, 317–322.

Zeilberger, D. and Bressoud,D.M. (1985). A proof of Andrews' q-Dyson conjecture, Discrete Math. 54, 201–224.

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My Mathematical Journey: The q-Dyson theorem