Schubert Polynomials and $k$-Schur functions

TL;DR

This paper explores the multiplication of Schubert polynomials by Schur functions through the lens of dual $k$-Schur functions, using quasisymmetric functions and poset structures to unify these concepts.

Contribution

It establishes a connection between Schubert vs. Schur multiplication and dual $k$-Schur functions via poset and graph structures, introducing new operators and frameworks.

Findings

Unified the Schubert vs. Schur problem with dual $k$-Schur functions.

Connected poset structures with affine Grassmannian order.

Defined new operators on dual $k$-Schur functions.

Abstract

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 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. On the other side, 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.