Non-crossing Cumulants of Type B

We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu. Lattices of non-crossing partitions (of type A, up to now) have played an important role in the combinatorics of free probability, primarily via the noncrossing cumulants of R. Speicher. Here we introduce the concept of noncrossing cumulant of type B; the inspiration for its definition is found by looking at an operation of “restricted convolution of multiplicative functions”, studied in parallel for functions on symmetric groups (in type A) and on hyperoctahedral groups (in type B). The non-crossing cumulants of type B live in an appropriate framework of “non-commutative probability space of type B”, and are closely related to a type B analogue for the R-transform of Voiculescu (which is the free probabilistic counterpart of the Fourier transform). By starting from a condition of “vanishing of mixed cumulants of type B”, we obtain an analogue of type B for the concept of free independence for random variables in a non-commutative probability space.