Monotonicity for generalized binomial coefficients and Jack positivity

TL;DR

This paper proves monotonicity of generalized binomial coefficients related to Jack and Schur polynomials and establishes new positivity inequalities in symmetric functions, extending classical inequalities like Newton and Muirhead.

Contribution

It introduces the monotonicity property for binomial coefficients associated with Jack and Schur polynomials and derives new positivity inequalities in symmetric functions.

Findings

Binomial coefficients are monotone for Jack and Schur polynomials

Established Schur positivity and Jack positivity inequalities

Extended classical inequalities like Newton and Muirhead

Abstract

Binomial formulas for Schur polynomials and Jack polynomials were studied by Lascoux in 1978, and Kaneko, Okounkov--Olshanski and Lassalle in the 1990s. We prove that the associated binomial coefficients are monotone and derive some symmetric function inequalities, in particular, a Schur positivity and Jack positivity result. These inequalities are similar to those studied by Newton, Muirhead, Gantmacher, Cuttler--Greene--Skandera, Sra and Khare--Tao.