Low-degree Boolean functions on $S_n$, with an application to isoperimetry

TL;DR

This paper characterizes Boolean functions on the symmetric group with Fourier concentration on large parts, showing they are close to unions of cosets, and proves sharp isoperimetric inequalities for subsets of permutations.

Contribution

It establishes a structural characterization of Boolean functions with Fourier concentration on large irreducible representations and derives sharp isoperimetric inequalities for subsets of the symmetric group.

Findings

Boolean functions with concentrated Fourier spectrum are close to unions of cosets

Derived asymptotically sharp edge-isoperimetric inequalities for the transposition graph

Confirmed a conjecture of Ben Efraim for large n relative to t

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 $n!/\textrm{poly}(n)$, 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.