Pieri's rule for flag manifolds and Schubert polynomials

TL;DR

This paper generalizes Pieri's rule from symmetric polynomials and Grassmannians to Schubert polynomials and flag manifolds, providing explicit formulas for multiplication in the cohomology ring.

Contribution

It introduces a geometric approach to compute structure constants for Schubert polynomials in flag manifolds, extending classical Pieri's rule.

Findings

Derived explicit formulas for Schubert polynomial multiplication

Connected structure constants to paths in Bruhat order

Extended Pieri's rule to flag manifolds

Abstract

We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary symmetric polynomial or a complete homogeneous symmetric polynomial. Thus, we generalize the classical Pieri's rule for symmetric polynomials/Grassmann varieties to Schubert polynomials/flag manifolds. Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, which we express in terms of paths in the Bruhat order on the symmetric group.