Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Double orthodontia formulas and Lascoux positivity

Linus Setiabrata, Avery St. Dizier

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.

Double orthodontia formulas and Lascoux positivity

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 . We also prove a curious positivity result: for vexillary permutations , the polynomial is a graded nonnegative sum of Lascoux polynomials. We conjecture that this positivity result holds for all . This conjecture would follow from a problem of independent interest regarding Lascoux positivity of certain products of Lascoux polynomials.

Paper Structure

This paper contains 4 sections, 27 theorems, 85 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Orthodontia and double Grothendieck polynomials
  4. Lascoux positivity

Key Result

Theorem 1.1

Let $D$ be a %-avoiding diagram with double orthodontic sequence $\mathbf K, \mathbf i, \mathbf j, \mathbf M$. Define When $D = D(w)$ is the Rothe diagram of a permutation, then $\mathscr G_D(\mathbf x, \mathbf y) = \mathfrak G_w(\mathbf x, \mathbf y)$.

Figures (6)

  • Figure 1: The five pipe dreams in $\mathop{\mathrm{PD}}\nolimits(1423)$.
  • Figure 2: The Rothe diagram for $w = 31542$.
  • Figure 3: A forbidden configuration in a %-avoiding diagram.
  • Figure 4: The double orthodontic sequence for $w = 31542$.
  • Figure 5: The Rothe diagram of $w = 68342751$ on the left, and of its sorting $w_\mathrm{sort} = 68234751$ on the right. In this example, $\sigma(w) = 231$.
  • ...and 1 more figures

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Lemma 2.1
  • Example 2.2
  • Theorem 2.3: weigandt21; see also km04, fk94
  • Definition 2.4
  • Example 2.5
  • ...and 55 more