Crystal Constructions in Number Theory
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.
I would like to thank the editors of this volume for giving me the opportunity to contribute. I am grateful to several people for helpful conversations and advice during the writing of this chapter, including Holley Friedlander, Paul E Gunnells, Dinakar Muthiah, and Manish Patnaik. During parts of the writing process, I was a postdoctoral fellow at the University of Alberta and a visiting assistant professor at the University of Massachusetts, Amherst, and I am grateful to both institutions. While at the University of Alberta, I was supported through Manish Patnaik’s Subbarao Professorship in number theory and an NSERC Discovery Grant. I also thank the referees for their insightful comments for the improvement of this chapter. In particular, I thank one of the referees for their comments on the connections to character theory and on Gauss sums.
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