Natural endomorphisms of shuffle algebras. (Q2842029)

As for any bialgebra, the vector space \(\mathrm{End}(\mathrm{Sh}(X))\) of all linear endomorphisms of a commutative shuffle bialgebra \(\mathrm{Sh}(X)\) of a graded set \(X\) is an associative algebra via the convolution product. In this paper the authors show that the dendriform algebra structure of \(\mathrm{Sh}(X)\) induces also such a structure on \(\mathrm{End}(\mathrm{Sh}(X))\) and the goal is then to investigate two natural dendriform subalgebras of \(\mathrm{End}(\mathrm{Sh}(X))\). The first one is defined by using graded permutations and is a natural extension of the Malvenuto-Reutenauer Hopf algebra [\textit{C. Malvenuto} and \textit{C. Reutenauer}, J. Algebra 177, No. 3, 967-982 (1995; Zbl 0838.05100)]. The other one, the dendriform descent algebra, is a subalgebra of the former. The authors study certain properties of these two algebras that are similar to those of the underlying shuffle algebras (namely, their bidendriform structures, freeness, and generators as well as bases and dimensions of their graded components). As an application they obtain a new proof of \textit{F. Chapoton}'s rigidity theorem for shuffle bialgebras [J. Pure Appl. Algebra 168, No. 1, 1-18 (2002; Zbl 0994.18006)].