Natural endomorphisms of shuffle algebras

TL;DR

This paper explores the structure of endomorphism algebras in shuffle algebras, extending classical algebraic tools to new contexts and revealing their foundational properties and relations.

Contribution

It introduces a natural endomorphism algebra for commutative shuffle algebras, extending the Malvenuto-Reutenauer algebra, and analyzes its structural properties.

Findings

Established freeness properties of the new algebra

Identified generators and bases of the algebra

Explored relations to the internal structure of shuffle algebras

Abstract

We focus in this text on the adaptation to the study of shuffles of the main combinatorial tool in the theory of free Lie algebras, namely the existence of a universal algebra of endomorphisms for tensor and other cocommutative Hopf algebras: the family of Solomon's descent algebras of type A. We show that there exists similarly a natural endomorphism algebra for commutative shuffle algebras, which is a natural extension of the Malvenuto-Reutenauer Hopf algebra of permutations, or algebra of free quasi-symmetric functions. We study this new algebra for its own, establish freeness properties, study its generators, bases, and also feature its relations to the internal structure of shuffle algebras.