Double orthodontia formulas and Lascoux positivity

TL;DR

This paper introduces a new formula for double Grothendieck polynomials using Magyar's orthodontia algorithm, proves a positivity property for vexillary permutations, and conjectures its general validity.

Contribution

It provides a novel combinatorial formula for double Grothendieck and Schubert polynomials and establishes a new positivity result for vexillary permutations.

Findings

New formula for double Grothendieck polynomials based on orthodontia algorithm.

Positivity result for vexillary permutations involving Lascoux polynomials.

Conjecture that the positivity extends to all permutations.

Abstract

We give a new formula for double Grothendieck polynomials based on Magyar's orthodontia algorithm for diagrams. Our formula implies a similar formula for double Schubert polynomials $\mathfrak S_w(\mathbf x;\mathbf y)$. We also prove a curious positivity result: for vexillary permutations $w\in S_n$, the polynomial $x_1^n\dots x_n^n \mathfrak S_w(x_n^{-1}, \dots, x_1^{-1}; 1,\dots,1)$ is a graded nonnegative sum of Lascoux polynomials. We conjecture that this positivity result holds for all $w\in S_n$. This conjecture would follow from a problem of independent interest regarding Lascoux positivity of certain products of Lascoux polynomials.