A product formula for certain Littlewood–Richardson coefficients for Jack and Macdonald polynomials

Abstract

Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley conjectured that if the Littlewood---Richardson coefficient for a triple of Schur polynomials is 1, then the corresponding coefficient for Jack polynomials can be expressed as a product of weighted hooks of the Young diagrams associated with the partitions indexing the coefficient. We prove a special case of this conjecture in which the partitions indexing the Littlewood---Richardson coefficient have at most three parts. We also show that this result extends to Macdonald polynomials.