  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 \lambda,
we denote by \lambda^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_{\lambda'}s_{\lambda^c}, where \lambda 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 \lambda. We prove the conjecture in many
cases.
