A Proof of the $G_2 $ Case of Macdonald’s Root System-Dyson Conjecture

Abstract

Dedicated to Joe Gillis on the occasion of his 75th birthday Key words, root systems, constant term identities, hypergeometric summation AMS(MOS) subject classifications. 31, 05A We will 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-)(1-)(1_)]m[(1-xY)(1-iz)(1-x Y-z) ] " C(m,n)= (3m + 3n)!(3n)!(2m)!(2n)! (2m + 3n) !(m + 2n) !(m + n)!m!n!n! This is the G2 case of Macdonald's Root System-Dyson conjecture (see [6, Conjecture 2.3, and (c), p. 994]; see also Morris [7]). Macdonald [6] showed how Selberg's integral [8] (see [1] for Aomoto's recent brilliant proof) implies his conjecture for all the so-called classical root systems. We will follow the same route and show how the G2 case follows from a corollary of *

A generalization of Selberg’s beta integral

“…Kadell [14] has previously proved these conjectures for all affine root systems of type 5(5^) and hence 5(2^), 5'(5 / ) v and SiDj). The Macdonald-Morris conjectures for R = G 2 have been proved by Habsieger [13] and Zeilberger [26]. See Garvan [8] for F 4 , Garvan and Gonnet [9] for 5(F 4 ) V , Zeilberger [27] for S(G 2 ) V and Opdam [20] for the q = 1 conjectures.…”

The importance of the Selberg integral

“…In a not dissimilar manner, D. Zeilberger [175] showed that the n = 3 case of the Morris integral (1.17) leads to the Macdonald conjecture for the exceptional root system G 2 . This result later found application in a study linking random matrix theory to number theoretical L-functions [97], see also Section 4.5.…”

“…Most other cases of (2.8) were proved on a case by case basis, often using methods based on q-integrals of Selberg type [51,65,66,73,87,156,175,176,177], but the three exceptional root systems E 6 , E 7 and E 8 refused to surrender until I. Cherednik gave a uniform proof for all reduced root systems based on his theory of double affine Hecke algebras [29,30,31,111].…”

A character sum for root system 𝐺₂

“…The form of the sum L in (1.9) is suggested by identifying the polynomial F(W) in (1.9) with (W-x/y)(W -y/z)(W -z/x), where x, v, z are the variables in the constant term identity for G2 in [11,Theorem,p. 880].…”