Natural Endomorphisms of Shuffle Algebras

Shuffles have a long history, starting with the probabilistic study of card shufflings in the first part of the 20th century by Borel, Hadamard, Poincaré and others. Their theory was revived in the 50’s, for various reasons. In topology, the combinatorics of (non commutative) shuffle products was the key to the definition of topological products such as the ones existing on cochain algebras and the cohomology groups of topological spaces. Commutative shuffle products were the key to the study of the homology of abelian groups and commutative algebras. In combinatorics and for the theory of iterated integrals, commutative shuffle products played a key role resulting in the global picture of the modern theory of free Lie algebras given in C. Reutenauer’s seminal Free Lie algebras [33]. The classical approach to shuffle algebras, as featured for example in Reutenauer’s book, focussed on Lie theoretical properties, that is on the enveloping algebra structure of tensor algebras: the shuffle product arises naturally in this framework by dualizing the Hopf algebra structure of the tensor algebra and many properties of shuffles can be derived from that particular approach. However, one can try to follow a different path, namely start directly from the combinatorics of shuffles, following the ideas originally developed by M.-P. Schützenberger [32]. A series of recent works by F. Chapoton, C. Malvenuto, C. Reutenauer, the second author of the present article, and others, provides many new tools to revisit the theory of shuffles. This is the purpose of the present article to put these tools to use. Concretely, we focus 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 [33, 27]. 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.