We prove a criterion of when the dual character $\chi _{D}(x)$ of the flagged Weyl module associated 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 Mészáros–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–Mészáros–St. Dizier and Hodges–Yong, respectively.

We study the back stable Schubert calculus of the infinite flag variety. Our main results are:

• – a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;

• – a novel definition of double and triple Stanley symmetric functions;

• – a proof of the positivity of double Edelman–Greene coefficients generalizing the results of Edelman–Greene and Lascoux–Schützenberger;

• – the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman–Greene insertion algorithm;

• – the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;

• – equivariant Pieri rules for the homology of the infinite Grassmannian;

• – homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.