A lift of Schur's Q-functions to the peak algebra

TL;DR

This paper introduces noncommutative Schur Q-functions within the peak algebra, providing new bases with positive rules and combinatorial properties, refining classical symmetric functions.

Contribution

It constructs a lift of Schur's Q-functions to the peak algebra, establishing new bases with desirable algebraic and combinatorial features.

Findings

Established a positive right-Pieri rule for the new basis

Derived a combinatorial expansion of the basis elements

Refined Schur's P-functions within the Stembridge algebra

Abstract

We construct a lift of Schur's Q-functions to the peak algebra of the symmetric group, called the noncommutative Schur Q-functions, and extract from them a new natural basis with several nice properties such as the positive right-Pieri rule, combinatorial expansion, etc. Dually, we get a basis of the Stembridge algebra of peak functions refining Schur's P-functions in a simple way.