Littlewood-Richardson coefficients for Jack polynomials

Yusra Naqvi (Sydney)

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.