CGS Conjecture for Jack polynomials (majorization via normalized Jack differences)

Establish the equivalence of the following four conditions for any two integer partitions λ and μ of the same size (|λ|=|μ|): (1) For all x∈[0,∞)^n, the normalized Jack difference P_λ(x;τ)/P_λ(1;τ) − P_μ(x;τ)/P_μ(1;τ) takes values in the cone ^R = {f/g : f,g ∈ ℝ_{≥0}[τ], g ≠ 0}; (2) For some fixed τ₀∈[0,∞], the same normalized Jack difference is nonnegative on [0,∞)^n; (3) For some fixed τ₀∈[0,∞], the same normalized Jack difference is nonnegative on (0,1)^n ∪ (1,∞)^n; (4) λ majorizes μ.

Background

This conjecture extends majorization-type evaluation positivity results from classical bases (monomials, elementary, power-sum, Schur) to Jack polynomials. It seeks to characterize majorization purely in terms of positivity properties of normalized Jack differences, generalizing results such as Muirhead’s inequality and Cuttler–Greene–Skandera’s inequalities to the Jack setting.

The paper proves many forward implications among the listed conditions but leaves the full equivalence as an explicit conjecture. It also shows partial cases (e.g., for two parts) and supporting evidence via integrals and semiring decompositions.