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POSITIVITY RESULTS ON RIBBON SCHUR FUNCTION DIFFERENCES
, 2007
"... There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those wh ..."
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There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that results from ordering such ribbons according to Schur positivity is essentially a product of two chains.
Maximal supports and Schur-positivity among connected skew shapes
, 2012
"... The Schur-positivity order on skew shapes is defined by B ≤ A if the difference sA − sB 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 sA−sB is that the support ..."
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The Schur-positivity order on skew shapes is defined by B ≤ A if the difference sA − sB 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 sA−sB 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 λ for which sλ appears in the Schur expansion of sB. 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.
Extended partial order and applications to tensor products
- Australasian Journal of Combinatorics
, 2014
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Multiplicity-free skew Schur functions with interval support
, 2010
"... Abstract: It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew-shape. We characterize skew Schur functions (and therefore the product of ..."
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Abstract: It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew-shape. We characterize skew Schur functions (and therefore the product of two Schur functions) which are multiplicity-free and the resulting Schur expansion runs over the whole interval of partitions, i.e., skew Schur functions having Littlewood-Richardson coefficients always equal to 1 over the full interval.
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 Schur equality: that sA and sB have ..."
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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 Schur equality: that sA and sB 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 sA contains that of sB, then the row overlap partitions of A are dominated by those of B, and again conjecture that the converse also holds. Our evi-dence 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
