Abstract
We prove that Boolean functions on$S_{n}$, whose Fourier transform is highly concentrated on irreducible representations indexed by partitions of$n$whose largest part has size at least$n-t$, are close to being unions of cosets of stabilizers of$t$-tuples. We also obtain an edge-isoperimetric inequality for the transposition graph on$S_{n}$which is asymptotically sharp for subsets of$S_{n}$of size$n!/\text{poly}(n)$, using eigenvalue techniques. We then combine these two results to obtain a sharp edge-isoperimetric inequality for subsets of$S_{n}$of size$(n-t)!$, where$n$is large compared to$t$, confirming a conjecture of Ben Efraim in these cases.
Highlights
This paper is part of a trilogy dealing with stability and ‘quasistability’ results concerning Boolean functions on the symmetric group, which are of ‘low complexity’.Let us begin with some notation and definitions which will enable us to present the Fourier-theoretic context of our results.c The Author(s) 2017
In [12], we proved that a Boolean function of expectation O(1/n), whose Fourier transform is highly concentrated on irreducible representations corresponding to the partitions (n) and (n − 1, 1), is close in structure to a union of 1-cosets
A Boolean function of expectation O(1/n), which is close to U1, has small symmetric difference with a union of 1-cosets. This is not true stability, as the Boolean function corresponding to T11 ∪ T22 is O(1/n2)-close to U1, but is not 1/(2n)-close to any dictatorship, whereas a Boolean function in U1 must be a dictatorship. We call it a ‘quasistability’ result. (In [13], on the other hand, we prove that a Boolean function of expectation bounded away from 0 and 1, which is close to U1, must be close in structure to a dictatorship; this is ‘genuine’ stability.)
Summary
This paper (together with [12] and [13]) is part of a trilogy dealing with stability and ‘quasistability’ results concerning Boolean functions on the symmetric group, which are (in a sense) of ‘low complexity’. In [12], we proved that a Boolean function of expectation O(1/n), whose Fourier transform is highly concentrated on irreducible representations corresponding to the partitions (n) and (n − 1, 1), is close in structure to a union of 1-cosets. We show that a Boolean function on Sn with expectation O(n−t ), whose Fourier transform is highly concentrated on irreducible representations indexed by partitions of n with first row of length at least n − t, is close in structure to a union of cosets of stabilizers of t-tuples. Put another way, a Boolean function of expectation O(n−t ), which is close to Ut (in Euclidean distance), has small symmetric difference with a union of t-cosets.
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