Abstracts

Symmetric functions and majorization inequalities: from Newton to Macdonald

Apoorva Khare, Indian Institute of Science

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.

I will then explain recent joint work with Hong Chen and Siddhartha Sahi, which explains how several of these inequalities are special cases of a “master” inequality characterizing majorization: using Jack, or even Macdonald, polynomials. We formulate this as a conjecture, and prove it (and the analogous weak-majorization conjecture) in two variables.