The Pieri rule for dual immaculate quasi-symmetric functions

TL;DR

This paper proves conjectures about the Pieri rules for dual immaculate quasi-symmetric functions, showing they are signed and multiplicity free, advancing the understanding of non-commutative symmetric functions.

Contribution

It establishes the validity of the conjectured signed, multiplicity free Pieri rules for dual immaculate quasi-symmetric functions.

Findings

Proved the left Pieri rule contains signs but is multiplicity free.

Confirmed the dual quasi-symmetric basis satisfies a signed, multiplicity free Pieri rule.

Enhanced the theoretical framework of non-commutative symmetric functions.

Abstract

The immaculate basis of the non-commutative symmetric functions was recently introduced by the first and third author to lift certain structures in the symmetric functions to the dual Hopf algebras of the non-commutative and quasi-symmetric functions. It was shown that immaculate basis satisfies a positive, multiplicity free right Pieri rule. It was conjectured that the left Pieri rule may contain signs but that it would be multiplicity free. Similarly, it was also conjectured that the dual quasi-symmetric basis would also satisfy a signed multiplicity free Pieri rule. We prove these two conjectures here.