KT Conjecture for Macdonald polynomials (weak majorization via shifted normalized Macdonald 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 Macdonald difference P_λ(x+1;q,t)/P_λ(1;q,t) − P_μ(x+1;q,t)/P_μ(1;q,t) takes values in the cone ℚ(q,t)_{≥0} = {f(q,t) : f(q,t)≥0 for q,t>1}; (2) For some fixed q₀,t₀>1, the same shifted normalized difference is nonnegative on [0,∞)^n; (3) λ weakly majorizes μ.

Background

This conjecture mirrors the Jack weak-majorization characterization in the Macdonald setting, with the evaluation shift x→x+1 and positivity interpreted through the Macdonald two-parameter cone. It connects weak majorization directly to normalized Macdonald differences.

The authors prove implications among the conditions and establish the conjecture for important classes (e.g., when μ is a vector of ones), leaving the full equivalence as an explicit conjecture.