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

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 (Gtilde,*) of unitized functions from the same incidence algebra, which satisfy a weaker condition of being "semi-multiplicative". The natural action of Gtilde 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 Gtilde 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 one-parameter interpolation between free and Boolean cumulants arising from work of Bozejko 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 Gtilde 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 of G in Gtilde turns out to be the dual of a natural bialgebra homomorphism from T onto Sym.

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