Kohnert polynomials

TL;DR

This paper introduces Kohnert polynomials derived from cell diagrams, generalizing Schubert polynomials and Demazure characters, and explores their stabilization to quasisymmetric functions with new bases.

Contribution

It defines Kohnert polynomials from cell diagrams, establishes their connection to known polynomial bases, and introduces two new Kohnert bases with conjectural positivity and stabilization properties.

Findings

Kohnert polynomials generalize Schubert and Demazure polynomials.

They stabilize to nonnegative quasisymmetric functions.

Two new bases are introduced, one conjecturally Schubert-positive.

Abstract

We associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts, with each choice giving rise to a combinatorially-defined basis of polynomials. These Kohnert bases provide a simultaneous generalization of Schubert polynomials and Demazure characters for the general linear group. Using the monomial and fundamental slide bases defined earlier by the authors, we show that Kohnert polynomials stabilize to quasisymmetric functions that are nonnegative on the fundamental basis for quasisymmetric functions. For initial applications, we define and study two new Kohnert bases. The elements of one basis are conjecturally Schubert-positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains the Schur functions.