Dyson conjecture

The Dyson conjecture states that the Laurent polynomial

${\displaystyle \prod _{1\leq i\neq j\leq n}(1-t_{i}/t_{j})^{a_{i}}}$

has constant term

${\displaystyle {\frac {(a_{1}+a_{2}+\cdots +a_{n})!}{a_{1}!a_{2}!\cdots a_{n}!}}.}$

The conjecture was first proved independently by Wilson (1962) and Gunson (1962). Good (1970) later found a short proof, by observing that the Laurent polynomials, and therefore their constant terms, satisfy the recursion relations

${\displaystyle F(a_{1},\dots ,a_{n})=\sum _{i=1}^{n}F(a_{1},\dots ,a_{i}-1,\dots ,a_{n}).}$

The case n = 3 of Dyson's conjecture follows from the Dixon identity.

Sills & Zeilberger (2006) and (Sills 2006) used a computer to find expressions for non-constant coefficients of Dyson's Laurent polynomial.

When all the values a_i are equal to β/2, the constant term in Dyson's conjecture is the value of Dyson's integral

${\displaystyle {\frac {1}{(2\pi )^{n}}}\int _{0}^{2\pi }\cdots \int _{0}^{2\pi }\prod _{1\leq j<k\leq n}|e^{i\theta _{j}}-e^{i\theta _{k}}|^{\beta }\,d\theta _{1}\cdots d\theta _{n}.}$

Dyson's integral is a special case of Selberg's integral after a change of variable and has value

${\displaystyle {\frac {\Gamma (1+\beta n/2)}{\Gamma (1+\beta /2)^{n}}}}$

which gives another proof of Dyson's conjecture in this special case.

Andrews (1975) found a q-analog of Dyson's conjecture, stating that the constant term of

${\displaystyle \prod _{1\leq i<j\leq n}\left({\frac {x_{i}}{x_{j}}};q\right)_{a_{i}}\left({\frac {qx_{j}}{x_{i}}};q\right)_{a_{j}}}$

is

${\displaystyle {\frac {(q;q)_{a_{1}+\cdots +a_{n}}}{(q;q)_{a_{1}}\cdots (q;q)_{a_{n}}}}.}$

Here (a;q)_n is the q-Pochhammer symbol. This conjecture reduces to Dyson's conjecture for q=1, and was proved by Zeilberger & Bressoud (1985), using a combinatorial approach inspired by previous work of Ira Gessel and Dominique Foata. A shorter proof, using formal Laurent series, was given in 2004 by Ira Gessel and Guoce Xin, and an even shorter proof, using a quantitative form, due to Karasev and Petrov, and independently to Lason, of Noga Alon's Combinatorial Nullstellensatz, was given in 2012 by Gyula Karolyi and Zoltan Lorant Nagy. The latter method was extended, in 2013, by Shalosh B. Ekhad and Doron Zeilberger to derive explicit expressions of any specific coefficient, not just the constant term, see http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html, for detailed references.

Macdonald (1982) extended the conjecture to arbitrary finite or affine root systems, with Dyson's original conjecture corresponding to the case of the A_n−1 root system and Andrews's conjecture corresponding to the affine A_n−1 root system. Macdonald reformulated these conjectures as conjectures about the norms of Macdonald polynomials. Macdonald's conjectures were proved by (Cherednik 1995) using doubly affine Hecke algebras.

Macdonald's form of Dyson's conjecture for root systems of type BC is closely related to Selberg's integral.

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