  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.
