Inequalities for finite group permutation modules
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- by Daniel Goldstein, Robert M. Guralnick and I. M. Isaacs PDF
- Trans. Amer. Math. Soc. 357 (2005), 4017-4042 Request permission
Abstract:
If $f$ is a nonzero complex-valued function defined on a finite abelian group $A$ and $\hat f$ is its Fourier transform, then $|\operatorname {supp}(f)||\operatorname {supp}({\hat f})| \ge |A|$, 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.References
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Additional Information
- Daniel Goldstein
- Affiliation: Center for Communications Research, 4320 Westerra Ct., San Diego, California 92121
- MR Author ID: 709300
- Email: dgoldste@ccrwest.org
- Robert M. Guralnick
- Affiliation: Department of Mathematics, University of Southern California, 1042 W. 36th Place, Los Angeles, California 90089
- MR Author ID: 78455
- 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
- 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 - © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2159698