Postnikov--Stanley polynomials are Lorentzian

TL;DR

This paper proves that Postnikov--Stanley polynomials, which generalize skew dual Schubert polynomials to arbitrary Weyl groups, are Lorentzian by linking them to Richardson varieties, thus connecting algebraic geometry with polynomial properties.

Contribution

It establishes that Postnikov--Stanley polynomials are Lorentzian and confirms their M-convex support, extending known results and resolving a recent conjecture.

Findings

Postnikov--Stanley polynomials are Lorentzian.

They are degree polynomials of Richardson varieties.

The result generalizes previous Lorentzian properties of dual Schubert polynomials.

Abstract

Postnikov--Stanley polynomials $D_u^w$ are a generalization of skew dual Schubert polynomials to the setting of arbitrary Weyl groups. We prove that Postnikov--Stanley polynomials are Lorentzian by showing that they are degree polynomials of Richardson varieties. Our result yields an interesting class of Lorentzian polynomials related to the geometry of Richardson varieties, generalizes the result that dual Schubert polynomials are Lorentzian (Huh--Matherne--M\'esz\'aros--St. Dizier 2022), and resolves the conjecture that Postnikov--Stanley polynomials have M-convex support (An--Tung--Zhang 2024).