Twisted descent algebras and the Solomon-Tits algebra

The purpose of the present article is to define and study a new class of descent algebras, called twisted descent algebras. These algebras are associated to the Barratt-Joyal theory of twisted bialgebras in the same way than classical descent algebras are associated to classical bialgebras. The theory of twisted descent algebras is a refinement of the theory of descent algebras of twisted algebras, as introduced in [17]. The formal properties of twisted descent algebras seem particularly meaningful in view of applications to discrete probabilities, to the geometry of Coxeter groups and buildings, and to symmetric group combinatorics. Let us survey briefly the classical theory, since most of the results that hold in that case have natural generalizations in the new setting. Recall first that classical shuffles and descent classes in the symmetric groups are one of the building blocks of modern algebraic combinatorics. Reutenauer’s