Schubert Polynomials and Elementary Symmetric Products

TL;DR

This paper investigates when Schubert polynomials factor into elementary symmetric polynomials, proposing a pattern avoidance criterion and providing partial proofs and obstructions related to permutation patterns.

Contribution

It introduces a conjecture linking permutation pattern avoidance to Schubert polynomial factorization and offers partial proofs and insights into this relationship.

Findings

Proved one direction of the conjecture.

Identified obstructions from permutation patterns.

Connected pattern avoidance to polynomial factorization.

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.