A quasi-stability result for dictatorships in S n

Abstract

We prove that Boolean functions on S n whose Fourier transform is highly concentrated on the first two irreducible representations of S n , are close to being unions of cosets of point-stabilizers. We use this to give a natural proof of a stability result on intersecting families of permutations, originally conjectured by Cameron and Ku [6], and first proved in [10]. We also use it to prove a ‘quasi-stability’ result for an edge-isoperimetric inequality in the transposition graph on S n , namely that subsets of S n with small edge-boundary in the transposition graph are close to being unions of cosets of point-stabilizers.

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Correspondence to David Ellis.

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Supported by the Canadian Friends of the Hebrew University/University of Toronto Permanent Endowment.

Supported in part by I.S.F. grant 0398246, and BSF grant 2010247.

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Ellis, D., Filmus, Y. & Friedgut, E. A quasi-stability result for dictatorships in S n . Combinatorica 35, 573–618 (2015). https://doi.org/10.1007/s00493-014-3027-1

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Mathematics Subject Classification (2000)

  • 05D99e