Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

Abstract

We consider the group (G, * ) of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where " * " denotes the convolution operation. We introduce a larger group ( G, * ) of unitized functions from the same incidence algebra, which satisfy a weaker semi-multiplicativity condition. The natural action of G on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of G in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of t-Boolean cumulants, a oneparameter interpolation between free and Boolean cumulants arising from work of Bożejko and Wysoczanski.It is known that the group G can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that G can also be identified as group of characters of a Hopf algebra T , which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion G ⊆ G turns out to be the dual of a natural bialgebra homomorphism from T onto Sym.