Schubert polynomials and patterns in permutations

TL;DR

This paper explores the support counts of Schubert polynomials, linking them to permutation patterns and lattice points in Newton polytopes, and improves existing bounds using dual flagged Weyl modules.

Contribution

It establishes a new lower bound for support counts of Schubert polynomials based on permutation patterns, extending the analysis within dual flagged Weyl modules.

Findings

Provides a lower bound for support counts in terms of pattern occurrences.

Connects support counts to lattice points in Newton polytopes.

Improves previous bounds for principal specializations of Schubert polynomials.

Abstract

This paper investigates the number of supports of the Schubert polynomial $\mathfrak{S}_w(x)$ indexed by a permutation $w$. This number also equals the number of lattice points in the Newton polytope of $\mathfrak{S}_w(x)$. We establish a lower bound for this number in terms of the occurrences of patterns in $w$. The analysis is carried out in the general framework of dual characters of flagged Weyl modules. Our result considerably improves the bounds for principal specializations of Schubert polynomials or dual flagged Weyl characters previously obtained by Weigandt, Gao, and M{\'e}sz{\'a}ros--St. Dizier--Tanjaya. Some problems and conjectures are discussed.