Quasisymmetric Schur functions

TL;DR

This paper introduces a new basis for quasisymmetric functions called quasisymmetric Schur functions, refining Schur functions and providing combinatorial formulas, expansions, and a Pieri rule, with extensions to Hall-Littlewood theory.

Contribution

The paper defines the quasisymmetric Schur basis, refines classical Schur functions, and develops combinatorial formulas and rules, extending to Hall-Littlewood polynomials.

Findings

New basis for quasisymmetric functions introduced

Derived expansions in monomial and fundamental bases

Established a Pieri rule refining the classical one

Abstract

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the $t$ parameter from Hall-Littlewood theory.