A positivity conjecture on the structure constants of shifted Jack functions

Abstract

We consider Jack polynomials J λ J_\lambda and their shifted analogue J λ # J^\#_\lambda . In 1989, Stanley conjectured that ⟨ J μ J ν , J λ ⟩ \langle J_\mu J_\nu , J_\lambda \rangle is a polynomial with nonnegative coefficients in the parameter α \alpha . In this note, we extend this conjecture to the case of shifted Jack polynomials.