# A Demazure crystal construction for Schubert polynomials

## TL;DR

This paper demonstrates that Schubert polynomials can be viewed as Demazure truncations of Stanley symmetric functions by establishing a Demazure crystal structure on key tableaux and linking it to type A crystal structures.

## Contribution

It introduces a Demazure crystal framework for Schubert polynomials, connecting them to Stanley symmetric functions through crystal structures on key tableaux.

## Key findings

- Schubert polynomials are Demazure truncations of Stanley symmetric functions.
- A Demazure crystal structure on key tableaux is established.
- Linking key tableaux with type A crystal structures enhances understanding of Schubert polynomials.

## 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.