A generating set of Solomonʹs descent algebra

Solomonʹs descent algebra is generated by sums of descent classes corresponding to certain hook shapes. This particularly implies that the ring of class functions of any finite symmetric group Sn is generated by the irreducible characters corresponding to certain hook partitions of n. As another consequence, a second generating set of Solomonʹs descent algebra (and of the ring of class functions of Sn) is obtained related to the major index of permutations.