A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions

TL;DR

This paper introduces new non-commutative bases that lift classical symmetric functions like Schur and Hall-Littlewood functions, preserving key properties and enabling positive expansions in related function spaces.

Contribution

It constructs novel non-commutative bases of symmetric functions and their duals, extending classical functions with similar properties in the non-commutative setting.

Findings

New basis of non-commutative symmetric functions with Schur function images

Dual basis of quasi-symmetric functions with positive expansions

Non-commutative lift of Hall-Littlewood functions with analogous properties

Abstract

We introduce a new basis of the non-commutative symmetric functions whose commutative images are Schur functions. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.