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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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