An Explicit Construction of Type A Demazure Atoms

TL;DR

This paper introduces a new explicit construction of Demazure atoms of type A using specializations of nonsymmetric Macdonald polynomials, providing a combinatorial interpretation that improves computation of key polynomials and their associated tableaux.

Contribution

It offers a novel explicit construction of Demazure atoms via nonsymmetric Macdonald polynomial specializations, enhancing combinatorial understanding and computational efficiency.

Findings

Demazure atoms can be derived from specialized nonsymmetric Macdonald polynomials.

The new approach accelerates computation of the right key for semi-standard Young tableaux.

Provides a new combinatorial description of key polynomials.

Abstract

Demazure characters of type A, which are equivalent to key polynomials, have been decomposed by Lascoux and Sch\"{u}tzenberger into standard bases. We prove that the resulting polynomials, which we call Demazure atoms, can be obtained from a certain specialization of nonsymmetric Macdonald polynomials. This combinatorial interpretation for Demazure atoms accelerates the computation of the right key associated to a semi-standard Young tableau. Utilizing a related construction, we provide a new combinatorial description for the key polynomials.