Quasisymmetric Schur functions

James Haglund,Kurt Luoto,Sarah Mason,Steph Van Willigenburg

doi:10.46298/dmtcs.3605

Abstract

We introduce a new basis for the algebra of quasisymmetric functions that naturally partitions Schur functions, called quasisymmetric Schur functions. We describe their expansion in terms of fundamental quasisymmetric functions and determine when a quasisymmetric Schur function is equal to a fundamental quasisymmetric function. We conclude by describing a Pieri rule for quasisymmetric Schur functions that naturally generalizes the Pieri rule for Schur functions. Nous étudions une nouvelle base des fonctions quasisymétriques, les fonctions de quasiSchur. Ces fonctions sont obtenues en spécialisant les fonctions de Macdonald dissymétrique. Nous décrivons les compositions que donne une simple fonction quasisymétriques. Nous décrivons aussi une règle par certaines fonctions de Schur.

Highlights

The Schur functions form an important basis for all symmetric functions, indexed by integer partitions
This paper presents a class of quasisymmetric functions that arises from the nonsymmetric Schur functions and decomposes the Schur functions
By combining the specialization of the nonsymmetric Macdonald polynomials to Demazure atoms mentioned in the introduction with the combinatorial formula for nonsymmetric Macdonald polynomials provided by Haglund, Haiman, and Loehr (3), we obtain a description of Demazure atoms in terms of new combinatorial objects which we describe

Summary

Introduction

The Schur functions form an important basis for all symmetric functions, indexed by integer partitions. They appear in the theory of Macdonald polynomials, (4), which are q, t-analogues of symmetric functions that were introduced to study problems from algebraic geometry. This paper presents a class of quasisymmetric functions that arises from the nonsymmetric Schur functions and decomposes the Schur functions. These functions, which we call quasisymmetric Schur functions, form a basis for the algebra of quasisymmetric functions.

Quasisymmetric and symmetric functions

A specialization of nonsymmetric Macdonald polynomials

Connection to fundamental quasisymmetric functions

A Pieri rule for quasisymmetric Schur functions