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Macdonald's Constant-Term Conjecture


Macdonald's constant term conjectures are related to root systems of Lie algebras (Macdonald 1982, Andrews 1986). They can be regarded as generalizations of Dyson's conjecture (Dyson 1962), its q-analog due to Andrews, and Mehta's conjecture (Mehta 2004). The simplest of these states that if R is a root system, then the constant term in product_(alpha in R)(1-e^alpha)^k, where k is a nonnegative integer, is product_(i=1)^(l)(kd_l; k), where the d_l are fixed integer parameters of the root system R corresponding to the fundamental invariants of the Weyl group W of R (Andrews 1986, p. 41).

Opdam (1989) proved the q=1 case for all root systems. The general conjecture had remained "almost proved" for some time, since the infinite families were accomplished by Zeilberger-Bressoud (A_n), Kadell (B_n, D_n), and Gustafson (BC_n, C_n), while the exceptional cases were done by Zeilberger and (independently) Habsieger (G_2), Zeilberger (G_2 dual), and Garvan and Gonnet (F_4 and F_4 dual), using Zeilberger's method. This left only the three root systems (E_6, E_7, E_8) which were infeasible to address using existing computers. In the meanwhile, however, Cherednik (1993) proved the constant term conjectures for all root systems using a methodology not dependent on classification.

A special case of the constant-term conjecture is given by the assertion that the constant term in

 product_(1<i!=j<=n)(1-(x_i)/(x_j))^k
(1)

is (nk)!/(k!)^n. Another special case asserts that the constant term in

 [product_(1<=i<=n)(x_i;q)_a(q/x_i;q)_a] 
 ×product_(1<=i<=j<=n)(x_ix_j;q)_b(q/(x_ix_j);q)_b((x_i)/(x_j);q)_b((qx_j)/(x_i);q)_b
(2)

is

 ((q;q)_(nb))/([(q;q)_b]^n)product_(1<=j<=n-1)((q;q)_(2a+2jb)(q;q)_(2jb))/((q;q)_(a+(n+j-1)n)(q;q)_(a+jb))
(3)

(Andrews 1986, p. 41).


See also

Dyson's Conjecture, Root System, Weyl Group

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References

Andrews, G. E. "The Macdonald Conjectures." §4.5 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 40-42, 1986.Cherednik, I. "The Macdonald Constant-Term Conjecture." Duke Math. J. 70, 165-177, 1993.Cherednik, I. "The Macdonald Constant-Term Conjecture." Internat. Math. Res. Not., No. 6, 165-177, 1993.Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. I." J. Math. Phys. 3, 140-156, 1962.Macdonald, I. G. "Affine Root Systems and Dedekind's eta-Function." Invent. Math. 15, 91-143, 1972.Macdonald, I. G. "Some Conjectures for Root Systems." SIAM J. Math. Anal. 13, 988-1007, 1982.Mehta, M. L. Random Matrices, 3rd ed. New York: Academic Press, 2004.Opdam, E. M. "Some Applications of Hypergeometric Shift Operators." Invent. Math. 98, 1-18, 1989.Stanton, D. "Sign Variations of the Macdonald Identities." SIAM J. Math. Anal. 17, 1454-1460, 1986.

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Macdonald's Constant-Term Conjecture

Cite this as:

Weisstein, Eric W. "Macdonald's Constant-Term Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MacdonaldsConstant-TermConjecture.html

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