A Demazure crystal construction for Schubert polynomials

Sami Assaf,Anne Schilling

doi:10.5802/alco.13

Abstract

Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by Morse and the second author in connection with Stanley symmetric functions.

Highlights

Schubert polynomials Sw were first introduced by Bernstein et al [6] as certain polynomial representatives of cohomology classes of Schubert cycles Xw in flag varieties
The Stanley symmetric functions Fw were introduced by Stanley [27] in the pursuit of enumerations of the reduced expressions of permutations, in particular of the long permutation w0
Stanley symmetric functions are the stable limit of Schubert polynomials [20, 21], precisely

Summary

Introduction

Schubert polynomials Sw were first introduced by Bernstein et al [6] as certain polynomial representatives of cohomology classes of Schubert cycles Xw in flag varieties. Demazure modules for the general linear group [9] are closely related to Schubert classes for the cohomology of the flag manifold In certain cases these modules are irreducible polynomial representations, and so the Demazure characters contain the Schur polynomials as a special case. Using a key tableaux interpretation for Demazure characters [3], Assaf [2] showed that the Edelman and Greene algorithm giving the Schur expansion of a Stanley symmetric function can be modified to a weak Edelman–Greene algorithm which gives the Demazure expansion of a Schubert polynomial. This gives a Demazure crystal structure for Schubert polynomials and shows that Schubert polynomials are a Demazure truncation of Stanley symmetric functions

Crystal structure on tableaux

Demazure crystal structure on key tableaux

Crystal structure for Stanley symmetric polynomials

Demazure crystal structure for Schubert polynomials