The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially the positivity of the multiplication of a dual $k$-Schur function by a Schur function.
@article{10_37236_4139,
author = {Carolina Benedetti and Nantel Bergeron},
title = {Schubert polynomials and {\(k\)-Schur} functions},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {4},
doi = {10.37236/4139},
zbl = {1302.05203},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4139/}
}
TY - JOUR
AU - Carolina Benedetti
AU - Nantel Bergeron
TI - Schubert polynomials and \(k\)-Schur functions
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/4139/
DO - 10.37236/4139
ID - 10_37236_4139
ER -
