Some positive differences of products of Schur functions

Abstract

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.

24 pages, 5 PostScript figures, multiple other figures use LaTeX picture environment