A positivity conjecture on the structure constants of shifted Jack functions

TL;DR

This paper extends Stanley's positivity conjecture from Jack polynomials to their shifted analogues, proposing that certain structure constants are polynomials with nonnegative coefficients in the parameter.

Contribution

It introduces a new positivity conjecture for shifted Jack polynomials, expanding the scope of Stanley's original conjecture.

Findings

Proposes a positivity conjecture for shifted Jack functions.

Extends Stanley's conjecture to a broader class of polynomials.

Provides theoretical groundwork for future proofs or computational verification.

Abstract

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