Necessary conditions for Schur-positivity

Abstract

In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference s A −s B of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Specifically, we prove that if s A −s B is Schur-positive, then certain row overlap partitions for A are dominated by those for B. In fact, our necessary conditions require a weaker condition than the Schur-positivity of s A −s B ; we require only that, when expanded in terms of Schur functions, the support of s A contains that of s B . In addition, we show that the row overlap condition is equivalent to a column overlap condition and to a condition on counts of rectangles fitting inside A and B. Our necessary conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for s A =s B , and we deduce a strengthening of their result as a special case.