Towards a polynomial basis of the algebra of peak quasisymmetric functions

TL;DR

This paper establishes a polynomial basis for the algebra of peak quasisymmetric functions over the rationals, demonstrating its freeness and providing a new structural understanding distinct from previous bases.

Contribution

It proves a structure theorem for PQSym over the integers, introduces a new polynomial basis over the rationals, and shows its freeness over a subring of symmetric functions.

Findings

Established a polynomial basis for PQSym over the rationals.

Proved the freeness of PQSym over its subring of symmetric functions.

Provided a new structural theorem for the algebra of peak quasisymmetric functions.

Abstract

Hazewinkel proved the Ditters conjecture that the algebra of quasisymmetric functions over the integers is free commutative by constructing a nice polynomial basis. In this paper we prove a structure theorem for the algebra of peak quasisymmetric functions (PQSym) over the integers. It provides a polynomial basis of PQSym over the rational field, different from Hsiao's basis, and implies the freeness of PQSym over its subring of symmetric functions spanned by Schur's Q-functions.