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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A proof of Andrews' $ q$-Dyson conjecture for $ n=4$


Author: Kevin W. J. Kadell
Journal: Trans. Amer. Math. Soc. 290 (1985), 127-144
MSC: Primary 33A15; Secondary 05A30
MathSciNet review: 787958
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Abstract: Andrews' $ q$-Dyson conjecture is that the constant term in a polynomial associated with the root system $ {A_{n - 1}}$ is equal to the $ q$-multinomial coefficient. Good used an identity to establish the case $ q = 1$, which was originally raised by Dyson. Andrews established his conjecture for $ n \leqslant 3$ and Macdonald proved it when $ {a_1} = {a_2} = \cdots = {a_n} = 1,2$ or $ \infty $ for all $ n \geqslant 2$. We use a $ q$-analog of Good's identity which involves a remainder term and linear algebra to establish the conjecture for $ n = 4$. The remainder term arises because of an essential problem with the $ q$-Dyson conjecture: the symmetry of the constant term. We give a number of conjectures related to the symmetry.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0787958-2
Article copyright: © Copyright 1985 American Mathematical Society