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

Understand

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.