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

We consider the group $(\mathcal{G},*)$ of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where ``$*$'' denotes the convolution operation. We introduce a larger group $(\widetilde{\mathcal{G}},*)$ of unitized functions from the same incidence algebra, which satisfy a weaker condition of being ``semi-multiplicative''. The natural action of $\widetilde{\mathcal{G}}$ 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 $\widetilde{\mathcal{G}}$ 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 one-parameter interpolation between free and Boolean cumulants arising from work of Bozejko and Wysoczanski. It is known that the group $\mathcal{G}$ can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that $\widetilde{\mathcal{G}}$ can also be identified as group of characters of a Hopf algebra $\mathcal{T}$, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion of $\mathcal{G}$ into $\widetilde{\mathcal{G}}$ turns out to be the dual of a natural bialgebra homomorphism from $\mathcal{T}$ onto Sym.