A stability result for balanced dictatorships in $S_{n}$

TL;DR

This paper demonstrates that Boolean functions on the symmetric group with Fourier concentration on the first two irreducible representations are structurally close to dictatorships, extending stability results in isoperimetric problems.

Contribution

It establishes a stability theorem for balanced Boolean functions on $S_{n}$ near dictatorships based on Fourier concentration, with implications for isoperimetric sets.

Findings

Functions close to dictatorships when Fourier is concentrated on first two irreducible representations.

Stability result applies when the function's expectation is bounded away from 0 and 1.

Contrasts with prior work where functions are close to unions of cosets of point-stabilizers.

Abstract

We prove that a balanced Boolean function on $S_{n}$ whose Fourier transform is highly concentrated on the first two irreducible representations of $S_{n}$, is close in structure to a dictatorship, a function which is determined by the image or pre-image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on $S_{n}$ generated by the transpositions. Our proof works in the case where the expectation of the function is bounded away from $0$ and $1$. In contrast, [Ellis, D., Filmus, Y., Friedgut, E., A quasi-stability result for dictatorships in $S_{n}$, Combinatorica 35 (2015), pp. 573-618] deals with Boolean functions of expectation O(1/n) whose Fourier transform is highly concentrated on the first two irreducible representations of $S_{n}$. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers.