What is Schur positivity and how common is it?

TL;DR

This paper explores Schur positivity, its connection to representation theory, and introduces a new probabilistic result on the likelihood of symmetric polynomials being Schur positive.

Contribution

It provides an explanation of Schur polynomials, their role in representation theory, and presents a novel probability result regarding Schur positivity of symmetric polynomials.

Findings

Derived the probability that a positive coefficient symmetric polynomial is Schur positive.

Clarified the role of Schur polynomials in the representation theory of GL(n).

Presented a new theoretical result on Schur positivity probability.

Abstract

This is a short note about Schur positivity. We introduce Schur polynomials and explain how they appear in the representation theory of the general linear group. We end with a new result of the author with F. Bergeron and V. Reiner that gives the probability that a homogeneous symmetric polynomial with positive coefficients is Schur positive.