Curtis Greene : Inequalities for families of symmetric functions

Abstract

We are interested in families of inequalities of the form f(X) >= g(X), where f(X) and g(X) are symmetric polynomials in X = (x_1,...,x_n) and the inequality must hold for all nonnegative substitutions of the variables. We will focus initially on inequalities involving well known combinatorial families (elementary, monomial, Schur, etc.). Far too much is known to permit a comprehensive survey, even within this limited scope, but there will be time to mention several interesting open problems and conjectures. The second part of the talk concerns a "positivity principle" that can be used to prove most (if not all) known symmetric function inequalities of this type, and apparently has not been studied before. Eventually, such proofs rest entirely on tableau combinatorics.