A positivity conjecture on the structure constants of shifted Jack functions

Abstract

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

9 pages, the main contribution is a conjecture with supporting numerical data; comments or solutions welcome!

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