A ``Fourier transform'' for multiplicative functions on non-crossing partitions
Free probability theory was introduced by D. Voiculescu about 15 years ago in connection with some old questions in the theory of operator algebras. Soon it turned out that `freeness' is not only an appropriate tool for dealing with these questions, but that it presents an abstract concept with an interesting structure of its own which has also many connections with other fields in mathematics and physics. One of these connections is with the combinatorics of non-crossing partitions. In the same way as classical probability theory can be described in combinatorial terms by multiplicative functions on all partitions of sets, free probability theory can be described by multiplicative functions on so-called non-crossing partitions. In the present work, the structure of the group of multiplicative functions on non-crossing partitions (with a canonical combinatorial convolution as group operation) is described by converting the convolution of multiplicative functions via a transform (called Fourier transform) into the multiplication of formal power series. On the combinatorial side, this gives an analogue for non-crossing partitions of a result for the lattice of all partitions of Doubilet, Rota and Stanley. On the probabilistic side, this result can be used to provide a new, totally combinatorial, proof of a theorem of Voiculescu concerning the distribution of the product of free random variables. Whereas the first part of the paper is totally combinatorial, the last part presents this result of Voiculescu and explains how the preceding work is related to it.
combinatorics of non-crossing partitions
scientific article; zbMATH DE number 1002031
scientific article