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

We prove that Boolean functions on , whose Fourier transform is highly concentrated on irreducible representations indexed by partitions of whose largest part has size at least , are close to being unions of cosets of stabilizers of -tuples. We also obtain an edge-isoperimetric inequality for the transposition graph on which is asymptotically sharp for subsets of of size , using eigenvalue techniques. We then combine these two results to obtain a sharp edge-isoperimetric inequality for subsets of of size , where is large compared to , confirming a conjecture of Ben Efraim in these cases.