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Every function on the Boolean cube $\{0,1\}^n$ has a unique presentations as a multilinear polynomial. This fails on the $(n,k)$ slice: for example, the multilinear polynomial $\sum_{i=1}^n x_i – k$ vanishes on the entire slice. An old result of Dunkl shows that functions on the slice can be presented uniquely as multilinear polynomials of degree at most $\min(k,n-k)$ satisfying an additional condition known as harmonicity: the sum of all partial derivatives is zero. An example of such a polynomial is $x_i – x_j$, and in fact the polynomials $x_i – x_j$ form a multiplicative basis for all harmonic multilinear polynomials.

We extend these results to the symmetric group and to the perfect matching scheme (the case of the multislice will be tackled in future work). In both cases, we extend the notion of harmonic to obtain a unique presentation theorem. In the case of the symmetric group, we also describe an explicit multiplicative basis.