Kohnert tableaux and a lifting of quasi-Schur functions

TL;DR

This paper introduces the quasi-key basis, a new polynomial basis that lifts quasi-Schur functions to the full polynomial ring, using Kohnert tableaux to describe positive expansions and stability properties.

Contribution

It defines the quasi-key basis, proves its relation to quasi-Schur polynomials, and develops Kohnert tableaux for combinatorial expansions and stability analysis.

Findings

Quasi-key basis contains quasi-Schur polynomials.

Key polynomials expand positively in quasi-key polynomials.

Kohnert tableaux provide combinatorial formulas for expansions.

Abstract

We introduce the quasi-key basis of the polynomial ring. We prove this basis contains the quasi-Schur polynomials of of Haglund, Luoto, Mason and van Willigenburg and that stable limits of quasi-key polynomials are quasi-Schur functions, thus giving a lifting of the quasi-Schur basis of quasisymmetric polynomials to the full polynomial ring. We introduce the combinatorial model of Kohnert tableaux, and use this model to prove that key polynomials expand positively in quasi-key polynomials which in turn expand positively in the fundamental slide polynomials introduced earlier by the authors. We give simple combinatorial formulas for these expansions in terms of Kohnert tableaux, lifting the parallel expansions of a Schur function into quasi-Schur functions into fundamental quasisymmetric functions. We further utilize Kohnert tableaux to find the precise point at which the fundamental slide expansion of a key polynomial stabilizes.