McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that $B\leq _s A$ if $s_A-s_B$ 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 $a$ or $(a+1)$ for $a\geq 2$. 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 $a$ being at the end, or on rows of length $(a+1)$ being evenly distributed.
@article{10_37236_7524,
author = {Foster Tom and Stephanie van Willigenburg},
title = {Necessary conditions for {Schur-maximality}},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {2},
doi = {10.37236/7524},
zbl = {1432.05117},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7524/}
}
TY - JOUR
AU - Foster Tom
AU - Stephanie van Willigenburg
TI - Necessary conditions for Schur-maximality
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/7524/
DO - 10.37236/7524
ID - 10_37236_7524
ER -
%0 Journal Article
%A Foster Tom
%A Stephanie van Willigenburg
%T Necessary conditions for Schur-maximality
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/7524/
%R 10.37236/7524
%F 10_37236_7524
Foster Tom; Stephanie van Willigenburg. Necessary conditions for Schur-maximality. The electronic journal of combinatorics, Tome 25 (2018) no. 2. doi: 10.37236/7524
