Littlewood-Richardson coefficients for Jack polynomials

Abstract:

Jack polynomials generalise several classical families of symmetric polynomials, including Schur polynomials. The coefficients which arise when the product of two Schur polynomials is expanded as a sum of Schur polynomials are the widely studied Littlewood-Richardson coefficients. This naturally leads to the question of finding suitable generalisations in the case of Jack polynomials. In 1989, Richard Stanley conjectured that whenever the Littlewood-Richardson coefficient for a triple of Schur polynomials is equal to 1, then the corresponding coefficient for Jack polynomials can be expressed as a product of weighted hooks of Young diagrams. In this talk, I will outline a proof of a special case of this conjecture, which uses a remarkable connection between Young tableaux and the integer points of certain polyhedral cones.