CGS conjecture for Jack polynomials

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. CGS conjecture for Jack polynomials

   Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$, and let $P_\lambda(x;\tau)$ denote the relevant Jack polynomial. Write

   $$\mathbb F_{\geqslant0}^{\mathbb R}=\left\{\frac{f}{g}:f,g\in\mathbb R_{\geqslant0}[\tau],\ g\neq0\right\}.$$

   Here $\lambda$ dominates $\mu$ when $\sum_{i=1}^k\lambda_i\geq\sum_{i=1}^k\mu_i$ for every $k$. CGS conjecture. The following are equivalent: $\lambda$ dominates $\mu$; there exists $\tau_0\in[0,\infty]$ such that

   $$\frac{P_\lambda(x;\tau_0)}{P_\lambda(\bm1;\tau_0)}-\frac{P_\mu(x;\tau_0)}{P_\mu(\bm1;\tau_0)}\geqslant0\quad\text{for all }x\in[0,\infty)^n;$$

   for every fixed $\tau_0\in[0,\infty]$ the same inequality holds; and

   $$\frac{P_\lambda(x;\tau)}{P_\lambda(\bm1;\tau)}-\frac{P_\mu(x;\tau)}{P_\mu(\bm1;\tau)}\in\mathbb F_{\geqslant0}^{\mathbb R}\quad\text{for all }x\in[0,\infty)^n.$$

   This is a proposed duality between dominance order and evaluation positivity for Jack polynomials; its resolution is not specified in the source.

   source: Hong Chen and Siddhartha Sahi, “Interpolation Polynomials, Binomial Coefficients, and Symmetric Function Inequalities”, arXiv:2403.02490 (2026).