Harmonicity and invariance on slices of the Boolean cube

In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree harmonic multilinear polynomials (a polynomial in x₁, … , x_n is harmonic if it is annihilated by ∑i=1n∂∂xi). Here we provide an alternative proof for general low-degree harmonic multilinear polynomials, with no constraints on the influences. We show that any real-valued harmonic multilinear polynomial on the slice whose degree is o(n) has approximately the same distribution under the slice and cube measures. Our proof is based on ideas and results from the representation theory of S_n, along with a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales. While such decompositions have been used in the past for stationary reversible Markov chains, our decomposition is applied in a non-stationary non-reversible setup. We also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof. Finally, we provide independent simple proofs for the known facts that (1) one cannot distinguish between the slice and the cube based on functions of o(n) coordinates and (2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is o(n).