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Transactions of the American Mathematical Society

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Non-crossing cumulants of type B

Authors: Philippe Biane, Frederick Goodman and Alexandru Nica
Journal: Trans. Amer. Math. Soc. 355 (2003), 2263-2303
MSC (2000): Primary 46L54, 05E15, 20F55
Published electronically: January 28, 2003
MathSciNet review: 1973990
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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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Additional Information

Philippe Biane
Affiliation: Département de Mathématiques et Applications de l’Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris, France

Frederick Goodman
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Alexandru Nica
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Received by editor(s): May 15, 2002
Received by editor(s) in revised form: September 18, 2002
Published electronically: January 28, 2003
Article copyright: © Copyright 2003 American Mathematical Society