Dual Schubert polynomials and their skew generalization, Postnikov-Stanley polynomials, possess deep geometric and algebraic properties. Dual Schubert polynomials, introduced by Bernstein, Gelfand, and Gelfand in 1973, represent the intersection degrees of Schubert varieties via Borel-Weil embeddings. In 2005, Postnikov and Stanley further showed that these polynomials form a dual basis to Schubert polynomials in the polynomial ring with infinitely many variables under the D-pairing, demonstrating that, despite their distinct geometric and combinatorial definitions, they encode the same information. Thus, research on dual Schubert polynomials sheds new light on classical problems in Schubert calculus.

In joint work with Serena An and Katherine Tung, we extend this perspective to the study of Postnikov-Stanley polynomials, proving that they represent the intersection degrees of Richardson varieties via Borel-Weil embeddings, and are therefore Lorentzian polynomials. Additionally, we provide an elegant and complete characterization of the supports of dual Schubert polynomials, which leads to a polynomial-time algorithm for determining whether a given term appears in a dual Schubert polynomial. I will also discuss open problems related to Postnikov-Stanley polynomials and their broader implications for Schubert calculus.