KT Conjecture for Jack polynomials (weak majorization via shifted normalized Jack differences)

Establish the equivalence of the following three conditions for any two integer partitions λ and μ: (1) For all x∈[0,∞)^n, the shifted normalized Jack difference P_λ(x+1;τ)/P_λ(1;τ) − P_μ(x+1;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 shifted normalized Jack difference is nonnegative on [0,∞)^n; (3) λ weakly majorizes μ.

Background

This conjecture upgrades known weak-majorization characterizations (previously established for Schur and other bases) to Jack polynomials with the evaluation shift x→x+1. It parallels and complements the majorization conjecture by using weak majorization and shifted arguments.

The paper proves chains of implications among these conditions and connects them to the majorization conjecture, but the full equivalence is left as an explicit conjecture.