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Transactions of the American Mathematical Society

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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
Published electronically: May 25, 2005
MathSciNet review: 2159698
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Abstract: If $f$ is a nonzero complex-valued function defined on a finite abelian group $A$ and $\hat f$ is its Fourier transform, then $\vert\operatorname{supp}(f)\vert\vert\operatorname{supp}({\hat f})\vert \ge \vert A\vert$, where $\operatorname{supp}(f)$ and $\operatorname{supp}({\hat f})$ are the supports of $f$ and $\hat f$. In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group $A$ is replaced by a transitive right $G$-set, where $G$ is an arbitrary finite group. We obtain stronger inequalities when the $G$-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

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

I. M. Isaacs
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706

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.

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