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    188宝金博页面版: A proof of the two parameter q-cases of the Macdonald - Morris constant term root system conjecture for S(F4) and S(F4)V via Zei

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    内容提示： J. Symbolic Computation (1992)14, 141-177A Proof of the Two Parameter q-Cases of theMacdonald - Morris Constant Term Root SystemConjecture forS(F4) and S(F4)vvia Zeilberger's MethodFRANK G . GARVANt AND GASTON H . GONNET 1tDepartment of Mathematics, University of Florida, Gainesville, FL 32611, U .S.A .IInformatik, ETH, Zurich, Switzerland 8092Doron Zeilberger has described a method for settling the q-case of the Macdonald-Morris root system constant term conjecture for any specific root system provided th...

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    J. Symbolic Computation (1992)14, 141-177A Proof of the Two Parameter q-Cases of theMacdonald - Morris Constant Term Root SystemConjecture forS(F4) and S(F4)vvia Zeilberger's MethodFRANK G . GARVANt AND GASTON H . GONNET 1tDepartment of Mathematics, University of Florida, Gainesville, FL 32611, U .S.A .IInformatik, ETH, Zurich, Switzerland 8092Doron Zeilberger has described a method for settling the q-case of the Macdonald-Morris root system constant term conjecture for any specific root system provided thereis sufficient computer time, memory space and some luck. He illustrated the methodby proving the S(G2)v case. His method involves finding and solving a linear systemof equations. We remove the element of luck by showing that it isalways possible toconstruct a triangular system . We apply the method to the so far open S(F4) and S(F4) vcases. A consequence of our triangularity result is that, in the equal parameter case, theMacdonald-Morris constant terms (for a fixed root system)form a q-hypergeometricsequence.1 . IntroductionIn 1982, Macdonald (1982) presented a collection of constant term conjecturesrelatingto root systems . Tire most general of these conjectures (Macdonald, 1982, Conj . 3 .3) iscast in the language of affine root systems S(R) and has the formC .T .11(q-x'; q°°)ka(qu°-e .x-°;qu-)ka= a certain explicit product . (1 .1)oER+Here C.T. means constant term in the Laurent polynomial in the x±a ; R is the underlyingroot system ; ko . are nonnegative integers satisfying ka. = k# whenever 11all =11,311 ; E,u0 are certain constant integers associated with the affine root system and (a ; q)k is thestandard q-notation(a)k = (a;q)k = (1 -a)(1 - aq)...(1 - aqk-1 ).The results of this paper were first announced in Macdonald's constant term conjectures for excep-tional root systems, Bulletin (new series) A . M . S ., 24 (1991), 343-347 .The research for this paper was done while the first author was a postdoctoral fellow at the Institutefor Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, and later asa Macquarie University Research Fellow at the School of Mathematics, Macquarie University, Sydney,NSW 2109, Australia .0747-7171/92/080141+37 $08 .00/0 © 1992 Academic Press Limited

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