Maximal Supports and Schur-positivity Among Connected Skew Shapes

Peter McNamara; Stephanie van Willigenburg

Maximal Supports and Schur-positivity Among Connected Skew Shapes

Peter McNamara, Stephanie van Willigenburg

Bucknell University

University of British Columbia

Research output: Contribution to journal › Article › peer-review

Abstract

The Schur-positivity order on skew shapes is defined by B ≤ A if the difference s A − s B 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 s A − s B 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 s λ appears in the Schur expansion of s B . 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.

Original language	American English
Journal	Default journal
Volume	33
State	Published - Aug 1 2012

Disciplines

Discrete Mathematics and Combinatorics

Access to Document

http://www.facstaff.bucknell.edu/pm040/Papers/maxsupport.pdf

Cite this

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title = "Maximal Supports and Schur-positivity Among Connected Skew Shapes",

abstract = " The Schur-positivity order on skew shapes is defined by B \≤ A if the difference s A \− s B 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 s A \− s B 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 s \λ appears in the Schur expansion of s B . 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. ",

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N2 - The Schur-positivity order on skew shapes is defined by B ≤ A if the difference s A − s B 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 s A − s B 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 s λ appears in the Schur expansion of s B . 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.

AB - The Schur-positivity order on skew shapes is defined by B ≤ A if the difference s A − s B 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 s A − s B 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 s λ appears in the Schur expansion of s B . 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.

UR - https://digitalcommons.bucknell.edu/fac_journ/117

UR - http://www.facstaff.bucknell.edu/pm040/Papers/maxsupport.pdf

M3 - Article

VL - 33

JO - Default journal

JF - Default journal

ER -

Maximal Supports and Schur-positivity Among Connected Skew Shapes

Abstract

Disciplines

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