Inequalities for finite group permutation modules

Authors:
Daniel Goldstein, Robert M. Guralnick and I. M. Isaacs

Journal:
Trans. Amer. Math. Soc. **357** (2005), 4017-4042

MSC (2000):
Primary 20B05; Secondary 20B15, 42A99

DOI:
https://doi.org/10.1090/S0002-9947-05-03927-9

Published electronically:
May 25, 2005

MathSciNet review:
2159698

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Abstract: If is a nonzero complex-valued function defined on a finite abelian group and is its Fourier transform, then , where and are the supports of and . In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group is replaced by a transitive right -set, where is an arbitrary finite group. We obtain stronger inequalities when the -set is primitive, and we determine the primitive groups for which equality holds. We also explore connections between inequalities of this type and a result of Chebotarëv on complex roots of unity, and we thereby obtain a new proof of Chebotarëv's theorem.

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Additional Information

**Daniel Goldstein**

Affiliation:
Center for Communications Research, 4320 Westerra Ct., San Diego, California 92121

Email:
dgoldste@ccrwest.org

**Robert M. Guralnick**

Affiliation:
Department of Mathematics, University of Southern California, 1042 W. 36th Place, Los Angeles, California 90089

Email:
guralnic@math.usc.edu

**I. M. Isaacs**

Affiliation:
Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706

Email:
isaacs@math.wisc.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03927-9

Received by editor(s):
October 24, 2003

Published electronically:
May 25, 2005

Additional Notes:
The research of the second author was partially supported by Grant DMS 0140578 of the U.S. NSF

The research of the third author was partially supported by the U.S. NSA

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.