  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.
