The shifted Poirier-Reutenauer algebra

TL;DR

This paper introduces a shifted version of the Poirier-Reutenauer algebra, connecting it with shifted Schensted correspondence, peak subalgebras, and Schur's P-functions, expanding algebraic combinatorics tools.

Contribution

It constructs a shifted Poirier-Reutenauer algebra as a higher lift of Schur's P-functions and explores its relations with peak subalgebras and Stembridge algebra.

Findings

Established the shifted Poirier-Reutenauer algebra as a right coideal subalgebra.

Linked the algebra with peak subalgebra and peak functions.

Connected the algebra to shifted Schensted correspondence and shifted Knuth equivalence.

Abstract

Based on the shifted Schensted correspondence and the shifted Knuth equivalence, a shifted analog of the Poirier-Reutenauer algebra as a higher lift of Schur's P-functions and a right coideal subalgebra of the Poirier-Reutenauer algebra is constructed. Its close relations with the peak subalgebra and the Stembridge algebra of peak functions are also uncovered.