I will give a gentle introduction to inequalities connecting symmetric polynomials and majorization. These have been studied by Maclaurin and Newton (1700s), Schlomilch (1800s), Gantmacher, Muirhead, Schur (1900s), and several others. Recently, Cuttler–Greene–Skandera and Sra (2010s) characterized majorization via inequalities involving Schur polynomials, elementary symmetric polynomials, and power sums. With Tao (2021), we analogously characterized weak majorization via Schur polynomials.

After discussing these and additional developments, I will speak about recent joint work with Hong Chen and Siddhartha Sahi, in which we conjecturally extend the above results, to characterize (weak) majorization via Jack and Macdonald polynomials. We prove these conjectures for all partitions with two parts.