The descent algebra of a graded bialgebra
The Solomon descent algebra of the symmetric group is naturally isomorphic to the convolution subalgebra of the endomorphism algebra of the tensor algebra (with its natural structure of bialgebra) generated by the graded projections. This construction is extended to each graded bialgebra (commutative or cocommutative); the resulting algebra is the descent algebra of the given graded bialgebra.
scientific article; zbMATH DE number 703234