Binary shuffle bases for quasi-symmetric functions

TL;DR

This paper develops bases for quasi-symmetric functions using binary shuffle products, linking them to multiple zeta values and extending to colored free quasi-symmetric functions, with implications for algebraic realizations.

Contribution

It introduces binary shuffle bases for quasi-symmetric functions and extends the construction to colored free quasi-symmetric functions, connecting to rational moulds.

Findings

Binary shuffle bases for quasi-symmetric functions constructed

Extension to algebra of colored free quasi-symmetric functions

Realization of algebra via rational moulds

Abstract

We construct bases of quasi-symmetric functions whose product rule is given by the shuffle of binary words, as for multiple zeta values in their integral representations, and then extend the construction to the algebra of free quasi-symmetric functions colored by positive integers. As a consequence, we show that the fractions introduced in [Guo and Xie, Ramanujan Jour. 25 (2011) 307-317] provide a realization of this algebra by rational moulds extending that of free quasi-symmetric functions given in [Chapoton et al., Int. Math. Res. Not. IMRN 2008, no. 9, Art. ID rnn018].