Schur-Positivity in a Square

Abstract

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem.  In this paper we study the Schur-positivity of a family of symmetric functions.  Given a partition $\nu$, we denote by $\nu^c$ its complement in a square partition $(m^m)$.   We conjecture a  Schur-positivity criterion  for symmetric functions of the form $s_{\mu'}s_{\mu^c}-s_{\nu'}s_{\nu^c}$, where $\nu$ is a partition of weight $|\mu|-1$ contained in $\mu$ and the complement of $\mu$ is taken in the same square partition as the complement of $\nu$. We prove the conjecture in many cases.