Maximal supports and Schur-positivity among connected skew shapes

TL;DR

This paper investigates the maximal elements in the Schur-positivity order among connected skew shapes, identifying key ribbon shapes and their supports to understand the structure of this ordering.

Contribution

It introduces a reduction to ribbon shapes for determining maximal connected skew shapes in the Schur-positivity order and explicitly characterizes their supports.

Findings

Support containment is a necessary condition for Schur-positivity differences.

Maximal connected skew shapes can be characterized via specific ribbon shapes.

Explicit support descriptions for these ribbon shapes are provided.

Abstract

The Schur-positivity order on skew shapes is defined by B \leq A if the difference s_A - s_B 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 s_A - s_B 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 lambda for which s_lambda appears in the Schur expansion of s_B. 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.