  The product $s_\mu s_\nu$ of two Schur functions is one of the most famous
examples of a Schur-positive function, i.e. a symmetric function which, when
written as a linear combination of Schur functions, has all positive
coefficients.
  We ask when expressions of the form $s_\lambda s_\rho - s_\mu s_\nu$ are
Schur-positive. This general question seems to be a difficult one, but a
conjecture of Fomin, Fulton, Li and Poon says that it is the case at least when
$\lambda$ and $\rho$ are obtained from $\mu$ and $\nu$ by redistributing the
parts of $\mu$ and $\nu$ in a specific, yet natural, way. We show that their
conjecture is true in several significant cases. We also formulate a skew-shape
extension of their conjecture, and prove several results which serve as
evidence in favor of this extension. Finally, we take a more global view by
studying two classes of partially ordered sets suggested by these questions.
