Multiplicative functions on the lattice of non-crossing partitions and free convolution

Abstract

The lattice of non-crossing partitions was introduced by Kreweras [Kre] in 1972 and examined further by Poupard [Pou] and Edelman [Edel, Ede2]. In the last time there has been an increased interest in this lattice, on one hand from the purely combinatorial point of view [SiU, EdS, Sire, BSS] and on the other hand from a (quantum) probabilistic point of view [GSS, Spel, Spe2, Spe3]. We became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures, the 'free' convolution introduced by Voiculescu [Voil, Voi2, Voi3]. This connection between the free convolution and the lattice of non-crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions. We shall work out this connection in the last part of this work. The intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other. Motivated by Voiculescu's formula for the R-series of free convolution we examine multiplicative functions on the lattice of non-crossing partitions and prove our main theorem, which is much in the spirit of Voiculescu's formula. We shall then use this main theorem for deriving some known results on non-crossing partitions in a unified way. This part of our work is purely combinatorial, the free convolution serves only as a motivation for our theorem. In the last section we shall switch to the probabilistie side and show how the lattice of non-crossing partitions determines the structure of the free convolution. Voiculescu's formula will then follow as an easy corollary of our main theorem. In this respect, we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools.