Row-strict quasisymmetric Schur functions

TL;DR

This paper introduces the row-strict quasisymmetric Schur function basis, a new combinatorial basis for quasisymmetric functions, and explores its relationships with existing bases and Schur polynomials, refining the omega transform operator.

Contribution

The paper presents a novel basis for quasisymmetric functions generated via row-strict tableau fillings, expanding the combinatorial and algebraic understanding of these functions.

Findings

Established the relationship between the new basis and existing quasisymmetric bases.

Connected the new basis to Schur polynomials.

Refined the omega transform operator based on these relationships.

Abstract

Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions called the row-strict quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through row-strict tableaux. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships.