The immaculate basis of the non-commutative symmetric functions

TL;DR

This paper introduces a novel basis for non-commutative symmetric functions that aligns with Schur functions in the commutative case, and develops a dual basis in quasi-symmetric functions with positive expansions and combinatorial Schur decompositions.

Contribution

It presents a new basis for non-commutative symmetric functions and a dual basis in quasi-symmetric functions with positive expansion properties.

Findings

New basis of non-commutative symmetric functions introduced

Dual basis in quasi-symmetric functions with positive expansion

Signed combinatorial formula for Schur function decomposition

Abstract

FPSAC 2013 Extended Abstract. We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula.