## 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 language | English (US) |
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Pages (from-to) | 151-175 |

Number of pages | 25 |

Journal | Journal of Number Theory |

Volume | 144 |

DOIs | |

State | Published - Nov 2014 |

Externally published | Yes |

## 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