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

DOI:
https://doi.org/10.1090/S0002-9947-03-03196-9

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.

**1.**D. Bessis. The dual braid monoid, preprint, January 2002 (arXiv:math.GR/0101158).**2.**P. Biane. Minimal factorizations of a cycle and central multiplicative functions on the infinite symmetric group, Journal of Combinatorial Theory Series A 76 (1996), 197-212. MR**97h:20003****3.**P. Biane. Some properties of crossings and partitions, Discrete Mathematics 175 (1997), 41-53. MR**98h:05020****4.**M. Bozejko and R. Speicher. -independent and symmetrized white noises, in Quantum Probability and Related Topics (L. Accardi editor), QP-PQ, VI, World Scientific, Singapore, (1991), 219-236.**5.**T. Brady. A partial order on the symmetric group and new 's for the braid groups, Advances in Mathematics 161 (2001), 20-40. MR**2002k:20066****6.**T. Brady and C. Watt. 's for Artin groups of finite type, preprint, February 2001 (arXiv:math.GR/0102078).**7.**P. Doubilet, G.-C. Rota, and R. Stanley. On the foundations of combinatorial theory (VI): The idea of generating function, Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, Lucien M. Le Cam et al. editors, University of California Press, 1972, 267-318. MR**53:7796****8.**P. de la Harpe. Topics in geometric group theory, University of Chicago Press, 2000. MR**2001i:20081****9.**J. E. Humphreys. Reflection groups and Coxeter groups, Cambridge University Press, 1990. MR**92h:20002****10.**B. Krawczyk and R. Speicher. Combinatorics of free cumulants, Journal of Combinatorial Theory Series A 90 (2000), 267-292. MR**2001f:46101****11.**G. Kreweras. Sur les partitions non croisées d'un cycle, Discrete Mathematics 1 (1972), 333-350. MR**46:8852****12.**A. Nica and R. Speicher. A ``Fourier transform'' for multiplicative functions on non-crossing partitions, Journal of Algebraic Combinatorics 6 (1997), 141-160. MR**98i:46070****13.**A. Nica and R. Speicher. On the multiplication of free -tuples of noncommutative random variables. With an Appendix by D. Voiculescu: Alternative proofs for the type II free Poisson variables and for the compression results. American Journal of Mathematics 118 (1996), 799-837. MR**98i:46069****14.**A. Nica and R. Speicher. The combinatorics of free probability, Preliminary version, December 1999. Lecture Notes of Centre Émile Borel, the Henri Poincaré Institute, Paris, France.**15.**A. Nica, D. Shlyakhtenko, and R. Speicher. Operator-valued distributions I. Characterizations of freeness, International Mathematics Research Notices 29 (2002), 1509-1538.**16.**V. Reiner. Non-crossing partitions for classical reflection groups, Discrete Mathematics 177 (1997), 195-222. MR**99f:06005****17.**R. Speicher. Multiplicative functions on the lattice of non-crossing partitions and free convolution, Mathematische Annalen 298 (1994), 611-628. MR**95h:05012****18.**R. Speicher. Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Memoirs of the Amer. Math. Soc. 132 (1998), no. 627. MR**98i:46071****19.**D. Voiculescu. Addition of certain noncommuting random variables, Journal of Functional Analysis 66 (1986), 323-346. MR**87j:46122****20.**D. Voiculescu, K. Dykema, and A. Nica. Free random variables, CRM Monograph Series, volume 1, American Mathematical Society, Providence, RI, 1992. MR**94c:46133**

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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

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

DOI:
https://doi.org/10.1090/S0002-9947-03-03196-9

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