Article - preprint
supercharacters, classical character theory, exponential sums, uncertainty principle, Fourier transform
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, 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.
© 2013 J.L. Brumbaugh, Madeleine Bulkow, Patrick S. Fleming, Luis Alberto Garcia, Stephan Ramon Garcia, Gizem Karaali, Matt Michal, Andrew P. Turner
Brumbaugh, J.L. '13; Bulkow, Madeleine '14; Fleming, Patrick S.; Garcia, Luis Alberto '14; Garcia, Stephan Ramon; Karaali, Gizem; Michal, Matt '15; and Turner, Andrew P. '14, "Supercharacters, Exponential Sums, and the Uncertainty Principle" (2013). Pomona Faculty Publications and Research. 146.