Zero-one dual characters of flagged Weyl modules

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{é}sz{á}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{é}sz{á}ros–St. Dizier and Hodges–Yong, respectively.