Crystal constructions in Number Theory

TL;DR

This paper explores the use of crystal graphs and Littelmann patterns to understand and construct prime power coefficients in Weyl group multiple Dirichlet series and metaplectic Whittaker functions, connecting algebraic combinatorics with number theory.

Contribution

It provides a comprehensive survey of combinatorial methods using crystal graphs for constructing key number-theoretic functions and highlights the role of crystal branching structures in these constructions.

Findings

Crystal graphs parameterized by Littelmann patterns effectively describe number-theoretic functions.

Branching structures of crystals reveal insights into prime power coefficients.

Combinatorial constructions facilitate access to open problems in number theory.

Abstract

Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics.