Low-degree Boolean functions on S_n, with an application to isoperimetry
Forum of Mathematics, Sigma
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We prove that Boolean functions on Sn, 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 Sn which is asymptotically sharp for subsets of Sn of size n!/poly(n), using eigenvalue techniques. We then combine these two results to obtain a sharp edge-isoperimetric inequality for subsets of Sn of size (n − t)!, where n is large compared to t, confirming a conjecture of Ben Efraim in these cases.
AuthorsELLIS, DC; Filmus, Y; Friedgut, E
- Mathematics