1 research outputs found
Necessary conditions for Schur-maximality
McNamara and Pylyavskyy conjectured precisely which connected skew shapes are
maximal in the Schur-positivity order, which says that if
is Schur-positive. Towards this, McNamara and van Willigenburg proved
that it suffices to study equitable ribbons, namely ribbons whose row lengths
are all of length or for . In this paper we confirm the
conjecture of McNamara and Pylyavskyy in all cases where the comparable
equitable ribbons form a chain. We also confirm a conjecture of McNamara and
van Willigenburg regarding which equitable ribbons in general are minimal.
Additionally, we establish two sufficient conditions for the difference of
two ribbons to be Schur-positive, which manifest as diagrammatic operations on
ribbons. We also deduce two necessary conditions for the difference of two
equitable ribbons to be Schur-positive that rely on rows of length being at
the end, or on rows of length being evenly distributed.Comment: 47 pages; final version to appear in Electron. J. Combi