A generalization of Edelman--Greene insertion for Schubert polynomials

TL;DR

This paper extends the Edelman--Greene insertion to provide explicit formulas for the Demazure character expansion of Schubert polynomials, connecting combinatorial algorithms with representation theory.

Contribution

It introduces a generalized Edelman--Greene correspondence that yields explicit formulas for Schubert polynomial expansions, utilizing dual equivalence techniques.

Findings

Derived explicit Demazure character formulas for Schubert polynomials

Connected Edelman--Greene insertion with positivity problems in algebraic combinatorics

Demonstrated applicability of dual equivalence methods to new positivity contexts

Abstract

Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman--Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial variation, but here we demonstrate how to extract explicit formulas from that machinery which may be applied to other positivity problems as well.