On the p-parts of Weyl group multiple Dirichlet series

We study the structure of $p$-parts of Weyl group multiple Dirichlet series. In particular, we extend results of Chinta, Friedberg, and Gunnells and show, in the stable case, that the $p$-parts of Chinta and Gunnells agree with those constructed using the crystal graph technique of Brubaker, Bump, and Friedberg. In this vein, we give an explicit recurrence relation on the coefficients of the $p$-parts, which allows us to describe the support of the $p$-parts and address the extent to which they are uniquely determined.