Positive differences of products of Schur functions

The product s_μ s_θ 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_λ s_ρ - s_μ s_ν are Schur-positive. We explain why this is a natural question, and why it seems to be a hard one. Conjectures of S.Fomin, W.Fulton, C.-K.Li and Y.-T.Poon concern the case when λ and ρ are obtained from μ and ν by certain natural operations on partitions. We study the combinatorics of these operations, and show that their conjectures are true for several significant families. We also formulate skew-shape extensions of their conjectures, and prove results which serve as evidence in favor of these extensions. Finally, we take a more global view by studying two classes of partially ordered sets suggested by the general framework.

This is joint work with Riccardo Biagioli, Peter McNamara, and Mercedes Rosas.