Inequalities for finite group permutation modules

Abstract

If f is a nonzero complex-valued function defined on a finite abelian group A and f is its Fourier transform, then | supp(f)∥supp(f)| ≥ |A|, where supp(f) and supp(f) are the supports of f and 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 Chebotarev on complex roots of unity, and we thereby obtain a new proof of Chebotarev's theorem.