A proof of Andrews' Dyson conjecture for
Author:
Kevin W. J. Kadell
Journal:
Trans. Amer. Math. Soc. 290 (1985), 127144
MSC:
Primary 33A15; Secondary 05A30
MathSciNet review:
787958
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Abstract: Andrews' Dyson conjecture is that the constant term in a polynomial associated with the root system is equal to the multinomial coefficient. Good used an identity to establish the case , which was originally raised by Dyson. Andrews established his conjecture for and Macdonald proved it when or for all . We use a analog of Good's identity which involves a remainder term and linear algebra to establish the conjecture for . The remainder term arises because of an essential problem with the Dyson conjecture: the symmetry of the constant term. We give a number of conjectures related to the symmetry.
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 [1]
 G. E. Andrews, Problems and prospects for basic hypergeometric functions, Theory and Application of Special Functions (R. A. Askey, ed.), Academic Press, New York, 1975, pp. 191224. MR 0399528 (53:3372)
 [2]
 , The theory of partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2 (G. C. Rota, ed.), AddisonWesley, Reading, Mass., 1976. MR 0557013 (58:27738)
 [3]
 , Notes on the Dyson conjecture, SIAM J. Math. Anal. 11 (1980), 787792. MR 586907 (82c:33002)
 [4]
 R. A. Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), 938951. MR 595822 (82e:33002)
 [5]
 , A beta integral associated with , SIAM J. Math. Anal. 13 (1982), 10081010. MR 674769 (84h:17006b)
 [6]
 W. N. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, Cambridge, 1935 (reprinted by Hafner, New York, 1964). MR 0185155 (32:2625)
 [7]
 D. E. Barton and C. L. Mallows, Some aspects of the random sequences, Ann. Math. Statist. 36 (1965), 236260. MR 0178488 (31:2745)
 [8]
 L. Biedenharn, W. Holman, III, and S. Milne, The invariant polynomials characterizing tensor operators having maximal null space, Adv. in Appl. Math. 1 (1980), 390472. MR 603138 (82k:81025)
 [9]
 L. Carlitz, Some formulas of F. H. Jackson, Monatsh. Math. 73 (1969), 193198. MR 0248035 (40:1290)
 [10]
 R. W. Carter, Simple groups of Lie type, Wiley, New York, 1972. MR 0407163 (53:10946)
 [11]
 A. L. Cauchy, Ouevres completes d'Augustin Cauchy, 1re Sér., Tome VIII, GauthierVillars, Paris, 1893.
 [12]
 A. C. Dixon, Summation of a certain series, Proc. London Math. Soc. (1) 35 (1903), 285289.
 [13]
 F. J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962), 140156. MR 0143556 (26:1111)
 [14]
 R. Evans, Character sum analogues of constant term identities for root systems, Israel J. Math. 46 (1983), 189196. MR 733348 (85c:11073)
 [15]
 C. F. Gauss, Werke, Vol. 3, Göttingen, 1866.
 [16]
 I. J. Good, Proofs of some 'binomial' identities by means of MacMahon's 'master theorem', Proc. Cambridge Philos. Soc. 58 (1962), 161162. MR 0137659 (25:1109b)
 [17]
 , Short proof of a conjecture of Dyson, J. Math. Phys. 11 (1970), 1884.
 [18]
 J. Gunson, Proof of a conjecture by Dyson in the statistical theory of energy levels, J. Math. Phys. 3 (1962), 752753. MR 0148401 (26:5908)
 [19]
 R. A. Gustafson and S. C. Milne, Schur functions, Good's identity, and hypergeometric series wellpoised in , Adv. in Math. 48 (1983), 177188. MR 700984 (84m:05013)
 [20]
 W. J. Holman, II, L. C. Biedenharn and J. D. Louck, On hypergeometric series wellpoised in , SIAM J. Math. Anal. 7 (1976), 529541. MR 0412504 (54:627)
 [21]
 E. C. Ihrig and M. E. H. Ismail, The cohomology of homogeneous spaces related to combinatorial identities (to appear).
 [22]
 F. H. Jackson, Certain identities, Quart. J. Math. 12 (1941), 167172. MR 0005963 (3:238d)
 [23]
 C. G. J. Jacobi, Gesammelte Werke, Vol. 1, Reimer, Berlin, 1881 (reprinted by Chelsea, New York, 1969).
 [24]
 K. W. J. Kadell, Weighted inversion numbers, restricted growth functions and standard Young tableaux, J. Combin. Theory Ser. A (to appear). MR 804866 (87c:05006)
 [25]
 , Andrews' Dyson conjecture II: Symmetry, Pacific J. Math. (to appear).
 [26]
 , A proof of some analogs of Selberg's integral for , preprint.
 [27]
 J. Lepowsky, Macdonaldtype identities, Adv. in Math. 27 (1978), 230234. MR 0554353 (58:27715)
 [28]
 J. Lepowsky and S. Milne, Lie algebraic approaches to classical partition identities, Adv. in Math. 29 (1978), 1559. MR 501091 (82f:17005)
 [29]
 J. D. Louck and L. C. Biedenharn, Canonical unit adjoint tensor operators in , J. Math. Phys. 11 (1970), 23682414. MR 0297237 (45:6295)
 [30]
 , On the structure of the canonical tensor operators in the unitary groups, III. Further developments of the boson polynomials and their implications, J. Math. Phys. 14 (1973), 13361357. MR 0342061 (49:6807)
 [31]
 I. G. Macdonald, Spherical functions on a adic Chevalley group, Bull. Amer. Math. Soc. 74 (1968), 520525. MR 0222089 (36:5141)
 [32]
 , Affine root systems and Dedekind's function, Invent. Math. 15 (1972), 91143. MR 0357528 (50:9996)
 [33]
 , Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), 9881007. MR 674768 (84h:17006a)
 [34]
 P. A. MacMahon, Combinatory analysis, Vol. 1, Cambridge Univ. Press, Cambridge, 1915 (reprinted by Chelsea, New York, 1960). MR 0141605 (25:5003)
 [35]
 S. C. Milne, Hypergeometric series wellpoised in and a generalization of Biedenharn's functions, Adv. in Math. 36 (1980), 169211. MR 574647 (81i:33010)
 [36]
 W. G. Morris, II, Constant term identities for finite and affine root systems: conjectures and theorems, Ph.D. Dissertation, University of WisconsinMadison, January 1982.
 [37]
 R. A. Proctor, Solution of two difficult combinatorial problems with linear algebra, Amer. Math. Monthly 89 (1982), 721734. MR 683197 (84f:05002)
 [38]
 A. Selberg, Bemerkninger om et multipelt integral, Norsk Mat. Tidsskr. 26 (1944), 7178. MR 0018287 (8:269b)
 [39]
 K. Wilson, Proof of a conjecture by Dyson, J. Math. Phys. 3 (1962), 10401043. MR 0144627 (26:2170)
 [40]
 D. Zeilberger, The algebra of linear partial difference operators and its applications, SIAM J. Math. Anal. 11 (1980), 919932. MR 595820 (81m:47047)
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DOI:
http://dx.doi.org/10.1090/S00029947198507879582
PII:
S 00029947(1985)07879582
Article copyright:
© Copyright 1985
American Mathematical Society
