The product s_μ s_ν of two Schur functions is one of the most famous examples of a Schur-positive function, i.e. a symmetric function which, when wri tten as a linear combination of Schur functions, has all positive coefficients. We ask when expressions of the form s_λ s_ρ - s_μ s_ν 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 λ and ρ are obtained from μ and ν by redistributing the parts of μ and ν 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.

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