Supercharacters, exponential sums, and the uncertainty principle

作者：Brumbaugh J L; Bulkow Madeleine; Fleming Patrick S; German Luis Alberto Garcia; Garcia Stephan Ramon^*; Karaali Gizem; Michal Matt; Turner Andrew P; Suh Hong

来源：Journal of Number Theory, 2014, 144: 151-175.

DOI：10.1016/j.jnt.2014.04.019

• 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

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.

• 出版日期2014-11