Kohnert Polynomials

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.

Keywords:

• Schubert polynomials
• Demazure characters
• key polynomials
• fundamental slide polynomials

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 14M15
• secondary 14N15
• 05E05

Acknowledgments

The authors thank Per Alexandersson, Nantel Bergeron, and Vic Reiner for helpful comments and illuminating discussions.

Additional information

Funding

S. Assaf was support by the Simons Foundation Collaboration Grant for Mathematicians (Award 524477).