Schubert Polynomials and Elementary Symmetric Products

TL;DR

The paper investigates when a Schubert polynomial $\mathfrak S_w$ factors as a product of elementary symmetric polynomials, formulating a pattern-avoidance characterization: $\mathfrak S_w=\prod_{t=1}^r e_{b_t}(x_1,\dots, x_{a_t})$ iff $w$ avoids $1432$, $1423$, $4132$, and $3142$. It develops a framework using Lehmer codes and pipe dreams to link pattern avoidance to factorization, proving one direction: avoidance of the four patterns ensures a decomposition because the Lehmer code constraints and a diagonal separation property force independent column contributions. It also identifies obstructions to the reverse direction by showing certain rectangular Lehmer-code configurations prevent such a factorization. Overall, the work highlights a concrete pattern-avoidance criterion that connects Schubert polynomials, elementary symmetric polynomials, and pipe-dream combinatorics, contributing to pattern-avoidance phenomena in algebraic combinatorics.

Abstract

We study the factorization of Schubert polynomials into elementary symmetric polynomials. We conjecture that this occurs when the permutation corresponding to the Schubert polynomial does not contain the patterns $1432$, $1423$, $4132$, and $3142$. We prove one direction of this and provide progress towards the second direction, including obstructions arising from permutations with a rectangular array of crosses in their bottom pipe dream. This characterization helps us identify new ties between elementary symmetric polynomials and Schubert polynomials. It contributes to the broader understanding of pattern avoidance phenomena in algebraic combinatorics.

Figures (3)

• Figure 1: Ladder Moves
• Figure 2: Two Pipe Dreams of the permutation 1432
• Figure 3: Pipe Dream of the permutation 431265

Theorems & Definitions (19)

• Example 1.1
• Example 1.2
• Definition 1.3
• Example 1.4
• Definition 1.5
• Example 1.6
• Definition 1.7
• Example 1.8
• Lemma 2.1
• proof