Necessary conditions for Schur-positivity
Peter R.W. McNamara
Bucknell University Department of Mathematics Lewisburg PA 17837 USA
DOI: 10.1007/s10801-007-0114-z
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.
Pages: 495–507
Keywords: keywords Schur function; skew Schur function; Schur-positivity; dominance order
Full Text: PDF
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