Hopf algebras and the logarithm of the $S$-transform in free probability

Let $k$ be a positive integer and let $\mathcal{G}_k$ denote the set of all joint distributions of $k$-tuples $(a_1,\ldots,a_k)$ in a noncommutative probability space $(\mathcal{A},\varphi)$ such that $\varphi(a_1)=\cdots=\varphi(a_k) = 1$. $\mathcal{G}_k$ is a group under the operation of the free multiplicative convolution $\boxtimes$. We identify $\bigl( \mathcal{G}_k, \boxtimes \bigr)$ as the group of characters of a certain Hopf algebra $\mathcal{Y}^{(k)}$. Then, by using the log map from characters to infinitesimal characters of $\mathcal{Y}^{(k)}$, we introduce a transform $LS_{\mu}$ for distributions $\mu \in \mathcal{G}_k$. $LS_{\mu}$ is a power series in $k$ noncommuting indeterminates $z_1, \ldots , z_k$; its coefficients can be computed from the coefficients of the $R$-transform of $\mu$ by using summations over chains in the lattices $NC(n)$ of noncrossing partitions. The $LS$-transform has the ``linearizing'' property that

$$LS_{\mu \boxtimes \nu} =LS_{\mu} +LS_{\nu}, \forall \mu , \nu \in \mathcal{G}_k$$     such that $$\mu \boxtimes \nu = \nu \boxtimes \mu .$$

In the particular case $k=1$ one has that ${\mathcal Y}^{(1)}$ is naturally isomorphic to the Hopf algebra Sym of symmetric functions and that the $LS$-transform is very closely related to the logarithm of the $S$-transform of Voiculescu by the formula

$$LS_{\mu} (z) = -z \log S_{\mu} (z), \forall \mu \in \mathcal{G}_1.$$

In this case the group $(\mathcal G_1, \boxtimes )$ can be identified as the group of characters of Sym, in such a way that the $S$-transform, its reciprocal $1/S$ and its logarithm $\log S$ relate in a natural sense to the sequences of complete, elementary and, respectively, power sum symmetric functions.