Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition $\nu$, we denote by $\nu^c$ its complement in a square partition $(m^m)$. We conjecture a Schur-positivity criterion for symmetric functions of the form $s_{\mu'}s_{\mu^c}-s_{\nu'}s_{\nu^c}$, where $\nu$ is a partition of weight $|\mu|-1$ contained in $\mu$ and the complement of $\mu$ is taken in the same square partition as the complement of $\nu$. We prove the conjecture in many cases.
@article{10_37236_3796,
author = {Cristina Ballantine and Rosa Orellana},
title = {Schur-positivity in a square},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {3},
doi = {10.37236/3796},
zbl = {1301.05355},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3796/}
}
TY - JOUR
AU - Cristina Ballantine
AU - Rosa Orellana
TI - Schur-positivity in a square
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/3796/
DO - 10.37236/3796
ID - 10_37236_3796
ER -
%0 Journal Article
%A Cristina Ballantine
%A Rosa Orellana
%T Schur-positivity in a square
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/3796/
%R 10.37236/3796
%F 10_37236_3796
Cristina Ballantine; Rosa Orellana. Schur-positivity in a square. The electronic journal of combinatorics, Tome 21 (2014) no. 3. doi: 10.37236/3796
