Schur-Positivity in a Square

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 ν, we denote by νc its complement in a square partition (mm). We conjecture a Schur-positivity criterion for symmetric functions of the form sμ′sμc − sν′sνc , where ν is a partition of weight |μ| − 1 contained in μ and the complement of μ is taken in the same square partition as the complement of ν. We prove the conjecture in many cases.