## Non-crossing cumulants of type B

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- by Philippe Biane, Frederick Goodman and Alexandru Nica
- Trans. Amer. Math. Soc.
**355**(2003), 2263-2303 - DOI: https://doi.org/10.1090/S0002-9947-03-03196-9
- Published electronically: 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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## Bibliographic Information

**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
**Alexandru Nica**- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- Email: anica@math.uwaterloo.ca
- Received by editor(s): May 15, 2002
- Received by editor(s) in revised form: September 18, 2002
- Published electronically: January 28, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**355**(2003), 2263-2303 - MSC (2000): Primary 46L54, 05E15, 20F55
- DOI: https://doi.org/10.1090/S0002-9947-03-03196-9
- MathSciNet review: 1973990