Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-commuting Variables

TL;DR

This paper introduces new bases for the Hopf algebra of quasisymmetric functions, refining existing symmetric powersum bases, and extends these concepts to non-commuting variables with explicit algebraic operations.

Contribution

It presents novel bases for quasisymmetric functions, including non-commuting variable analogs, with explicit algebraic structures and change-of-basis rules.

Findings

New bases refined from symmetric powersum basis

Explicit algebraic structures with shuffle product and deconcatenate coproduct

Change of basis rules between quasisymmetric powersum and fundamental bases

Abstract

We introduce new bases for the Hopf algebra of quasisymmetric functions that refine the symmetric powersum basis. These bases are expanded in terms of quasisymmetric monomial functions by using fillings of matrices. We define the analog of these bases in quasisymmetric functions of non-commuting variables. Our new bases have a (shifted) shuffle product and a deconcatenate coproduct. Finally, we describe a change of basis rule from the quasisymmetric powersum basis to the quasisymmetric fundamental basis.