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

doi:10.1016/j.jnt.2014.04.019

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 journal › Article › peer-review

19 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 language	English (US)
Pages (from-to)	151-175
Number of pages	25
Journal	Journal of Number Theory
Volume	144
DOIs	https://doi.org/10.1016/j.jnt.2014.04.019
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

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10.1016/j.jnt.2014.04.019

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title = "Supercharacters, exponential sums, and the uncertainty principle",

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. Andr{\'e}. 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.",

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",

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

note = "Funding Information: S.R. Garcia was partially supported by NSF Grants DMS-1001614 and DMS-1265973 . G. Karaali was partially supported by a NSA Young Investigator Award ( NSA Grant # H98230-11-1-0186 ). P.S. Fleming, S.R. Garcia, and G. Karaali were partially supported by the American Institute of Mathematics (AIM) and NSF Grant DMS-0901523 via the REUF Program. We also gratefully acknowledge the support of the Fletcher Jones Foundation and Pomona College's SURP Program . ",

year = "2014",

month = nov,

doi = "10.1016/j.jnt.2014.04.019",

language = "English (US)",

volume = "144",

pages = "151--175",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

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AU - Brumbaugh, J. L.

AU - Bulkow, Madeleine

AU - Fleming, Patrick S.

AU - Garcia German, Luis Alberto

AU - Garcia, Stephan Ramon

AU - Karaali, Gizem

AU - Michal, Matt

AU - Turner, Andrew P.

AU - Suh, Hong

N1 - Funding Information: S.R. Garcia was partially supported by NSF Grants DMS-1001614 and DMS-1265973 . G. Karaali was partially supported by a NSA Young Investigator Award ( NSA Grant # H98230-11-1-0186 ). P.S. Fleming, S.R. Garcia, and G. Karaali were partially supported by the American Institute of Mathematics (AIM) and NSF Grant DMS-0901523 via the REUF Program. We also gratefully acknowledge the support of the Fletcher Jones Foundation and Pomona College's SURP Program .

PY - 2014/11

Y1 - 2014/11

N2 - 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.

AB - 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.

KW - Circulant matrix

KW - Conjugacy class

KW - DCT

KW - DFT

KW - Discrete Fourier transform

KW - Discrete cosine transform

KW - Fourier transform

KW - Gauss sum

KW - Gaussian period

KW - Heilbronn sum

KW - Kloosterman sum

KW - Ramanujan sum

KW - Supercharacter

KW - Superclass

KW - Symmetric group

KW - Uncertainty principle

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VL - 144

SP - 151

EP - 175

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

Supercharacters, exponential sums, and the uncertainty principle

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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