Towards a Parsimonious Calculation of Jack Polynomials

We prove the following new result concerning coefficients of expansions of Jack polynomials in monomial and elementary symmetric functions. Up to normalisation, we show that those coefficients are unchanged upon removing the first row or column of the Ferrers diagrams of the partitions used for indexing, or upon reflecting the Ferrers diagrams in a sufficiently large grid. Our results exhibit a strong duality between expansion of Jack polynomials in monomial and elementary symmetric functions; and provide an important step towards the efficient calculation of Jack polynomials through their parsimonious expansion in those symmetric functions