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 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.
Highlights
A fundamental problem in algebraic combinatorics is to find combinatorial rules for certain properties of a given combinatorial Hopf algebra
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
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
Summary
A fundamental problem in algebraic combinatorics is to find combinatorial rules for certain properties of a given combinatorial Hopf algebra. In a series of two papers we plan to give a positive rule (along the lines of [3]) for the multiplication of dual k-Schur with a Schur function and relate this to the Schubert vs Schur problem. This is done by an in-depth study of the affine strong Bruhat graph. In this paper we will cover part (I) together with some related work and a combinatorial explicit embedding of the Schubert vs Schur problem into the dual k-Schur problem This is done by inclusion of the chains of the grassmannian-Bruhat order into the affine strong Bruhat graph.
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