Determining relations among symmetric functions continues to be a topic of considerable interest in algebraic combinatorics. We will focus on relations among Schur functions and their most classical generalization, the skew Schur functions.
To any skew shape A, we can associate the skew Schur function sA. We can then order the set of skew shapes by saying that A ≤ B if sA - sB is Schur-positive, i.e., when expanded in the basis of Schur functions, all the coefficients are non-negative. We call the resulting poset the Schur-positivity poset on skew shapes. While much recent work on Schur-positivity can be formulated in terms of the Schur-positivity poset, a complete understanding of the poset is presently well out of reach. After giving the necessary background and introducing the Schur-positivity poset, we will present necessary conditions on the shapes of A and B for A ≤ B. We will conclude with broad open questions in the area.