Maximal supports and Schur-positivity among connected skew shapes

The Schur-positivity order on skew shapes is defined by B≤AB≤A if the difference sA−sBsA−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−sBsA−sB is that the support of BB is contained in that of AA, where the support of BB is defined to be the set of partitions λλ for which sλsλ appears in the Schur expansion of sBsB. 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.

► Maximal connected skew shapes in Schur-positivity and support orders are ribbons.
► We determine the maximal connected skew shapes in the support containment order.
► We explicitly determine the support for maximal ribbon shapes.