Majorization via positivity of Jack and Macdonald polynomial differences
Hong Chen, Apoorva Khare, Siddhartha Sahi
TL;DR
This work extends majorization-type inequalities from classical symmetric polynomials to Jack and Macdonald polynomials, formulating conjectures that relate partition dominance to positivity of normalized differences. The authors establish two-variable results and develop a framework based on positivity cones and Muirhead semirings, enabling containment- and evaluation-positivity characterizations to be unified across Jack and Macdonald families. They prove several implications and provide concrete two-variable examples, with Kadell integrals furnishing supporting evidence. The study advances duality perspectives between partition orders and positivity notions, and lays groundwork for Macdonald extensions, potentially broadening the reach of majorization inequalities in symmetric function theory.
Abstract
Majorization inequalities have a long history, going back to Maclaurin and Newton. They were recently studied for several families of symmetric functions, including by Cuttler--Greene--Skandera (2011), Sra (2016), Khare--Tao (2021), McSwiggen--Novak (2022), and Chen--Sahi (2024+) among others. Here we extend the inequalities by these authors to Jack and Macdonald polynomials, and obtain conjectural characterizations of majorization and of weak majorization of the underlying partitions. We prove these characterizations for two variables. In fact, we upgrade -- and prove in the above cases -- the characterization of majorization, to containment of Jack and Macdonald differences lying in the Muirhead semiring.