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Recent Trends in Algebraic Combinatorics pp 333–362Cite as

Crystal Constructions in Number Theory

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Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 16))

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.

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Notes

  1. 1.

    In type D, the picture is more complex. An analogous construction gives a result slightly different from the one expected from the p-part. The phenomenon of (symmetric) multiple leaners accounts for this discrepancy; see Sect. 4.3.4 for details.

  2. 2.

    Note that in type B, we have the coefficient 1 instead of \(q^{a_{i,j}}.\) This discrepancy by a factor of q also present when comparing (47) with (46) or (48) can be eliminated with a change of variables in the polynomial \(P_{\lambda }({\mathbf {x}}).\)

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Acknowledgements

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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Authors and Affiliations

  1. Kavli Institute for the Physics and Mathematics of the Universe, Kashiwa, Chiba, Japan

    Anna Puskás

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  1. Anna Puskás
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Correspondence to Anna Puskás .

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Editors and Affiliations

  1. Mathematical Sciences Research Institute , Berkeley, CA, USA

    Hélène Barcelo

  2. Department of Mathematics, Pomona College, Claremont, CA, USA

    Gizem Karaali

  3. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    Rosa Orellana

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Puskás, A. (2019). Crystal Constructions in Number Theory. In: Barcelo, H., Karaali, G., Orellana, R. (eds) Recent Trends in Algebraic Combinatorics. Association for Women in Mathematics Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-030-05141-9_10

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