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ON MATRICES ARISING IN FINITE FIELD HYPERGEOMETRIC FUNCTIONS

Part of: Exponential sums and character sums Polynomials and matrices

Published online by Cambridge University Press:  22 April 2024

SATOSHI KUMABE Open the ORCID record for SATOSHI KUMABE [Opens in a new window]  and
HASAN SAAD Open the ORCID record for HASAN SAAD [Opens in a new window]
SATOSHI KUMABE*
Affiliation:
Joint Graduate School of Mathematics for Innovation, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka 819-0395, Japan
HASAN SAAD
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA e-mail: hs7gy@virginia.edu
*
e-mail: kuma511ssk@gmail.com
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Abstract

Lehmer [‘On certain character matrices’, Pacific J. Math. 6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math. 10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.

Keywords

cyclotomic matricescharacteristic polynomialfinite field hypergeometric functions

MSC classification

Primary: 11C20: Matrices, determinants
Secondary: 11L05: Gauss and Kloosterman sums; generalizations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 6
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was supported by JSPS KAKENHI Grant Number JP22KJ2477 and WISE program (MEXT) at Kyushu University. The second author thanks Ken Ono for providing research support with the Thomas Jefferson Fund and the NSF Grant (DMS-2002265 and DMS-2055118).

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