Skew row-strict quasisymmetric Schur functions

TL;DR

This paper introduces Young row-strict quasisymmetric Schur functions, explores their combinatorial properties, and establishes their algebraic relations, including skew functions, decompositions, and multiplication rules within the quasisymmetric functions framework.

Contribution

It defines Young row-strict quasisymmetric Schur functions, connects them to existing functions, and develops their algebraic and combinatorial properties, including skew functions and multiplication rules.

Findings

Defined Young row-strict quasisymmetric Schur functions.

Proved equivalence of algebraic and combinatorial definitions.

Established multiplication rules with Schur functions.

Abstract

Mason and Remmel introduced a basis for quasisymmetric functions known as the row-strict quasisymmetric Schur functions. This basis is generated combinatorially by fillings of composition diagrams that are analogous to the row-strict tableaux that generate Schur functions. We introduce a modification known as Young row-strict quasisymmetric Schur functions, which are generated by row-strict Young composition fillings. After discussing basic combinatorial properties of these functions, we define a skew Young row-strict quasisymmetric Schur function using the Hopf algebra of quasisymmetric functions and then prove this is equivalent to a combinatorial description. We also provide a decomposition of the skew Young row-strict quasisymmetric Schur functions into a sum of Gessel's fundamental quasisymmetric functions and prove a multiplication rule for the product of a Young row-strict quasisymmetric Schur function and a Schur function.