Supercharacters, exponential sums, and the uncertainty principle

J. L. Brumbaugh, Madeleine Bulkow, Patrick S. Fleming, Luis Alberto Garcia German, Stephan Ramon Garcia, Gizem Karaali, Matt Michal, Andrew P. Turner, Hong Suh

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

Abstract

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. André. 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.

Original languageEnglish (US)
Pages (from-to)151-175
Number of pages25
JournalJournal of Number Theory
Volume144
DOIs
StatePublished - Nov 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Keywords

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

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