Quasisymmetric divided difference operators and polynomial bases

TL;DR

This paper extends the use of quasisymmetric divided difference operators to define and analyze polynomial bases like key polynomials and Demazure atoms, revealing their connections to existing bases and exploring their properties.

Contribution

It introduces natural analogs of key polynomials and Demazure atoms using Hivert's operator and relates them to known bases like fundamental slide polynomials.

Findings

Key polynomials and Demazure atoms have analogs via Hivert's operator.

The resulting bases are related to fundamental slide polynomials and fundamental particle basis.

Structure constants for the fundamental particle basis are provided.

Abstract

The key polynomials, the Demazure atoms, the Schubert polynomials, and even the Schur functions can be defined using divided difference operator. In 2000, Hivert introduced a quasisymmetric analog of the divided difference operator. In particular, replacing it in a natural way in the definition of the Schur functions gives Gessel's fundamental basis. This paper is our attempt to apply the same methods to define the remaining bases and study the results. In particular, we show both the key polynomials and Demazure atoms have natural analogs using Hivert's operator and that the resulting bases occur independently and defined by other means in the work of Assaf and Searles, as the fundemental slide polynomials and the fundamental particle basis respectively. We further explore properties of these two bases, including giving the structure constants for the fundamental particle basis.