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

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.