CGS and Muirhead conjectures for Macdonald polynomials (majorization via normalized Macdonald differences)

Establish the equivalence of the following five conditions for any two integer partitions λ and μ with |λ|=|μ|: (1) For all q,t, the normalized Macdonald difference P_λ(x;q,t)/P_λ(1;q,t) − P_μ(x;q,t)/P_μ(1;q,t) lies in the Muirhead semiring ℳ_S(ℚ(q,t)); (2) For all x∈[0,∞)^n, the same normalized Macdonald difference takes values in the cone ℚ(q,t)_{≥0} = {f(q,t) : f(q,t)≥0 for q,t>1}; (3) For some fixed q₀,t₀>1, the difference is nonnegative on [0,∞)^n; (4) For some fixed q₀,t₀>1, the difference is nonnegative on (0,1)^n ∪ (1,∞)^n; (5) λ majorizes μ.

Background

This conjecture transfers the Jack positivity and semiring characterizations of majorization to Macdonald polynomials, introducing the two-parameter positivity cone and Muirhead semiring. It aims to unify evaluation positivity, semiring decompositions, and majorization for Macdonald polynomials.

The authors prove chains of implications and settle the equivalence in several cases (e.g., μ being all ones and for n=2), providing substantial evidence while leaving the full equivalence as an open conjecture.