Quasi-symmetric functions

Let 2 denote the Leibniz-Hopf algebra, which also turns up as the Solomon descent algebra, and the algebra of noncommutative symmetric functions. As an algebra Z = Z < Z(1),Z(2), ...>, the free associative algebra over the integers in countably many indeterminates. The co-algebra structure is given by mu(Z(n),) = Sigma(i=0)(n), Z(i) x Z(n-i), Z(0)= 2. Let M be the graded dual of Z. This is the algebra of quasi-symmetric functions. The Ditters conjecture (1972), says that this algebra is a free commutative algebra over the integers. This was proved in [13]. In this paper I give an outline of the proof and discuss a number of consequences and related matters.