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A proof of the \(G_ 2\) case of Macdonald’s root System-Dyson conjecture. (English) Zbl 0643.05004

We prove the following theorem.
Theorem. Let m and n be integers and x, y and z commuting indeterminates; then the constant term of the Laurent polynomial \[ F(x,y,z)=[(1- \frac{x}{y})(1-\frac{y}{z})(1-\frac{z}{x})]^ m[(1-\frac{xy}{z^ 2})(1- \frac{xz}{y^ 2})(1-\frac{yz}{x^ 2})]^ n \]
\[ [(1-\frac{y}{x})(1- \frac{z}{y}(1-\frac{x}{z})]^ m[(1-\frac{z^ 2}{xy})(1-\frac{y^ 2}{xz})(1-\frac{x^ 2}{yz})]^ n \] is \[ C(m,n)=(3m+3n)!(3n)!(2m)!(2n)!/((2m+3n)!(m+2n)!(m+n)!m!n!n!). \] This is the \(G_ 2\) case of Macdonald’s Root System-Dyson conjecture (see I. G. Macdonald [ibid. 13, 988-1007 (1982; Zbl 0498.17006) Conjecture 2.3, and (c), p. 994]; see also W. Morris [Constant term identities for finite and affine root systems, Ph. D. thesis, Univ. Wisconsin, Madison (1982)]).
Macdonald [loc. cit.] showed how Selberg’s integral (see K. Aomoto [Jacobi polynomials associated with Selberg integrals (to appear)] for his recent brilliant proof) implies his conjecture for all the so-called classical root systems. We follow the same route and show how the \(G_ 2\) case follows from a corollary of Selberg’s integral that is due to Morris [loc. cit., p. 94].
After the first version of this paper was written, I was kindly informed by Dominique Foata that Laurent Habsieger [C. R. Acad. Sci., Paris, Sér. I 303, 211-213 (1986; Zbl 0598.05006)] has independently and simultaneously obtained the results of this paper.
Reviewer: D.Zeilberger

MSC:

05A15 Exact enumeration problems, generating functions
17B20 Simple, semisimple, reductive (super)algebras
33C05 Classical hypergeometric functions, \({}_2F_1\)
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