Maximal Supports and Schur-positivity Among Connected Skew Shapes
The Schur-positivity order on skew shapes is defined by B≤A if the difference sA−sB is Schur-positive. It is an open problem to determine those connected skew shapes that are maximal with respect to this ordering. A strong necessary condition for the Schur-positivity of sA−sB is that the support of B is contained in that of A, where the support of B is defined to be the set of partitions λ for which sλ appears in the Schur expansion of sB. We show that to determine the maximal connected skew shapes in the Schur-positivity order and this support containment order, it suffices to consider a special class of ribbon shapes. We explicitly determine the support for these ribbon shapes, thereby determining the maximal connected skew shapes in the support containment order.
European Journal of Combinatorics
McNamara, Peter and van Willigenburg, Stephanie. "Maximal Supports and Schur-positivity Among Connected Skew Shapes." European Journal of Combinatorics (2012) : 1190-1206.