On the crystal graph description of the stable Weyl group multiple Dirichlet series

TL;DR

This paper establishes a new parametrization of Weyl group elements for certain Lie algebras and compares two descriptions of Weyl group multiple Dirichlet series, enhancing understanding in algebraic and number theoretic contexts.

Contribution

It introduces a coordinate-free parametrization of Weyl group elements and compares crystal graph and Lie-theoretic descriptions of Weyl group multiple Dirichlet series.

Findings

Parametrization for Weyl group elements in semisimple Lie algebras.

Comparison between crystal graph and Lie-theoretic descriptions.

Clarification of the structure of Weyl group multiple Dirichlet series.

Abstract

For a semisimple Lie algebra admitting a good enumeration, we prove a parametrization for the elements in its Weyl group. As an application, we give a coordinate-free comparison between the crystal graph description (when it is known) and the Lie-theoretic description of the Weyl group multiple Dirichlet series in the stable range.