On the p-parts of Weyl group multiple Dirichlet series

TL;DR

This paper investigates the structure of p-parts of Weyl group multiple Dirichlet series, extending previous results and providing explicit recurrence relations to understand their support and uniqueness.

Contribution

It extends prior work by showing p-parts agree with crystal graph constructions in the stable case and introduces recurrence relations for their coefficients.

Findings

p-parts agree with crystal graph constructions in the stable case

recurrence relations for coefficients are explicitly given

support and uniqueness of p-parts are characterized

Abstract

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.