Double Schubert polynomials for the classical groups

TL;DR

This paper introduces a new family of double Schubert polynomials for classical Lie groups of types B, C, and D, extending previous single-variable polynomials to an equivariant setting with stability and positivity properties.

Contribution

It defines and studies double Schubert polynomials for classical groups, generalizing earlier single-variable polynomials and connecting them to factorial Schur Q- and P-functions.

Findings

Polynomials represent Schubert classes in equivariant cohomology.

Polynomials satisfy a stability property.

Explicit expressions relate to factorial Schur Q- and P-functions.

Abstract

For each infinite series of the classical Lie groups of type B,C or D, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's Q- or P-functions defined earlier by Ivanov.