A generalization of Edelman–Greene insertion for Schubert polynomials

Sami H Assaf

doi:10.5802/alco.160

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.

Highlights

Schur functions, the ubiquitous basis for symmetric functions with deep connections to representation theory and geometry, may be regarded as the generating functions for standard Young tableaux
Edelman and Greene [10] established a bijective correspondence between reduced words and ordered pairs of Young tableaux of the same partition shape such that the left is increasing with reduced reading word and the right is standard
Through this correspondence they proved Stanley’s conjecture and, gave an explicit formula for the Schur expansion as the number of such left tableaux that can appear in the correspondence

Summary

Introduction

The ubiquitous basis for symmetric functions with deep connections to representation theory and geometry, may be regarded as the generating functions for standard Young tableaux. Edelman and Greene [10] established a bijective correspondence between reduced words and ordered pairs of Young tableaux of the same partition shape such that the left is increasing with reduced reading word and the right is standard Through this correspondence they proved Stanley’s conjecture and, gave an explicit formula for the Schur expansion as the number of such left tableaux that can appear in the correspondence. Schubert polynomials were introduced by Lascoux and Schützenberger [14] as polynomial representatives of Schubert classes for the cohomology of the flag manifold with nice algebraic and combinatorial properties They can be defined as the generating polynomials of reduced words [5, 7], and in the stable limit, they become the Stanley symmetric functions [16]. Demazure characters, key polynomials, RSK, Edelman–Greene insertion, reduced words

Generating functions for reduced words

Equivalence relations

Positive expansions

Insertion algorithms