Comparing skew Schur functions: a quasisymmetric perspective

Reiner, Shaw, and van Willigenburg showed that if two skew Schur functions sA and sB are equal, then the skew shapes A and B must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than skew Schur equality: that sA and 5B have the same support when expanded in the fundamental quasisymmetric basis F. Surprisingly, there is significant evidence supporting a conjecture that the converse is also true. In fact, we work in terms of inequalities, showing that if the F-support of 5A contains that of 5B, then the row overlap partitions of A are dominated by those of B, and again conjecture that the converse also holds. Our evidence in favor of these conjectures includes their consistency with a complete determination of all F support containment relations for F-multiplicity-free skew Schur functions. We conclude with a consideration of how some other quasisymmetric bases fit into our framework.