A Note on Jing and Li's Type B Quasisymetric Schur Functions

TL;DR

This paper proves Jing and Li's conjecture that type B quasisymmetric Schur functions expand positively into peak functions, refines their combinatorial model, and clarifies their algebraic properties and limitations.

Contribution

The paper confirms the positivity conjecture for type B quasisymmetric Schur functions and provides explicit expansions in multiple bases, enhancing understanding of their structure.

Findings

Confirmed positive, integral, unitriangular expansion into peak functions.

Provided explicit expansions in monomial, fundamental, and peak bases.

Showed these functions are not quasisymmetric Schur, Young quasisymmetric Schur, or dual immaculate positive.

Abstract

In 2015, Jing and Li defined type B quasisymmetric Schur functions and conjectured that these functions have a positive, integral and unitriangular expansion into peak functions. We prove this conjecture, and refine their combinatorial model to give explicit expansions in monomial, fundamental and peak bases. We also show that these functions are not quasisymmetric Schur, Young quasisymmetric Schur or dual immaculate positive, and do not have a positive multiplication rule.