Transactions of the American Mathematical Society

Author(s): Philippe Biane; Frederick Goodman; Alexandru Nica
Journal: Trans. Amer. Math. Soc. 355 (2003), 2263-2303.
MSC (2000): Primary 46L54, 05E15, 20F55
Posted: January 28, 2003
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Abstract: 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 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.

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Philippe Biane
Affiliation: Département de Mathématiques et Applications de l'Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris, France
Email: Philippe.Biane@ens.fr

Frederick Goodman
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email: goodman@math.uiowa.edu

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