McNamara, Peter R.W.

Necessary conditions for Schur-positivity

J. Algebr. Comb. 28(4), 495-507 (2008)
DOI: 10.1007/s10801-007-0114-z


Summary: 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.


keywords Schur function, skew Schur function, Schur-positivity, dominance order