Supercharacters, exponential sums, and the uncertainty principle

JOURNAL OF NUMBER THEORY

卷:144

Article

Brumbaugh, J. L.³  Bulkow, Madeleine¹  Fleming, Patrick S.²  German, Luis Alberto Garcia³  Garcia, Stephan Ramon³  Michal, Matt⁴  Turner, Andrew P.⁵  Suh, Hong³ 

[1] Scripps Coll, Dept Math, Claremont, CA 91711 USA

[2] South Dakota Sch Mines & Technol, Math & Comp Sci Dept, Rapid City, SD 57701 USA

[3] Pomona Coll, Dept Math, Claremont, CA 91711 USA

[4] Inst Math Sci, Claremont, CA 91711 USA

[5] Harvey Mudd Coll, Dept Math, Claremont, CA 91711 USA

关键词: Supercharacter;    Conjugacy class;    Superclass;    Circulant matrix;    Discrete Fourier transform;    DFT;    Discrete cosine transform;    DCT;    Fourier transform;    Gauss sum;    Gaussian period;    Ramanujan sum;    Heilbronn sum;    Kloosterman sum;    Symmetric group;    Uncertainty principle;   

DOI  :  10.1016/j.jnt.2014.04.019

来源: Elsevier

【 摘 要 】

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. Andre. We study supercharacter theories on (Z/nZ)(d) induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, Heilbronn, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free