Zero-one dual characters of flagged Weyl modules

TL;DR

This paper characterizes when the dual character of flagged Weyl modules, including special cases like Schubert and key polynomials, has coefficients only 0 or 1, confirming a conjecture and unifying previous criteria.

Contribution

It provides a new criterion for zero-one dual characters of flagged Weyl modules, settling a conjecture and unifying existing results for Schubert and key polynomials.

Findings

Established a criterion for zero-one dual characters of flagged Weyl modules.

Unified proof for zero-one conditions of Schubert and key polynomials.

Confirmed a conjecture by M{é}sz{á}ros--St. Dizier--Tanjaya.

Abstract

We prove a criterion of when the dual character $\chi_{D}(x)$ of the flagged Weyl module associated to a diagram $D$ in the grid $[n]\times [n]$ is zero-one, that is, the coefficients of monomials in $\chi_{D}(x)$ are either 0 or 1. This settles a conjecture proposed by M{\'e}sz{\'a}ros--St. Dizier--Tanjaya. Since Schubert polynomials and key polynomials occur as special cases of dual flagged Weyl characters, our approach provides a new and unified proof of known criteria for zero-one Schubert/key polynomials due to Fink--M{\'e}sz{\'a}ros--St. Dizier and Hodges--Yong, respectively.