Quasi-Symmetric Functions

Abstract

Let Z 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 < Z1, Z2,... >, the free associative algebra over the integers in countably many indeterminates. The co-algebra structure is given by \(\mu({Z_n})=\sum\nolimits_{i = 0}^n{{Z_i}}\otimes{Z_{n-i}}\), Z 0 = 1. 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.