Non-crossing cumulants of type B

Details Non-crossing cumulants of type B Non-crossing cumulants of type B

2002-06-17

arxiv.org/abs/math/0206167v1

Mathematics

Operator Algebras

48 pages, latex

Scientific paper

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 non-crossing cumulants of R. Speicher. Here we introduce the concept of {\em non-crossing 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.

Affiliated with

Also associated with

No associations

Rating

Non-crossing cumulants of type B does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Non-crossing cumulants of type B, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Non-crossing cumulants of type B will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-603343