Harmonic Polynomials on Perfect Matchings

We show that functions over perfect matchings of complete graphs admit unique presentations as harmonic polynomials annihilated by certain differential operators. Moreover, we give a concrete description of these harmonic polynomials by computing the unique harmonic presentation of the standard basis of Specht polynomials. At the core of these results is a class of incidence matrices that we call the matching inclusion matrices. The algebraic combinatorics of these matrices are related to Jack polynomials, which leads us to some elegant formulas for particular weighted sums of Jack characters for arbitrary α. Along the way, we prove a perhaps new combinatorial identity related to Jack characters that equates the product of the top row of α-upper hook lengths of a shape λ to a weighted sum of so-called tableau transversals of λ.