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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. D. Bessis. The dual braid monoid, preprint, January 2002 (arXiv:math.GR/0101158).
  • 2. Philippe Biane, Minimal factorizations of a cycle and central multiplicative functions on the infinite symmetric group, J. Combin. Theory Ser. A 76 (1996), no. 2, 197–212. MR 1416014, 10.1006/jcta.1996.0101
  • 3. Philippe Biane, Some properties of crossings and partitions, Discrete Math. 175 (1997), no. 1-3, 41–53. MR 1475837, 10.1016/S0012-365X(96)00139-2
  • 4. M. Bozejko and R. Speicher. $\psi$-independent and symmetrized white noises, in Quantum Probability and Related Topics (L. Accardi editor), QP-PQ, VI, World Scientific, Singapore, (1991), 219-236.
  • 5. Thomas Brady, A partial order on the symmetric group and new 𝐾(𝜋,1)’s for the braid groups, Adv. Math. 161 (2001), no. 1, 20–40. MR 1857934, 10.1006/aima.2001.1986
  • 6. T. Brady and C. Watt. $K( \pi , 1)$'s for Artin groups of finite type, preprint, February 2001 (arXiv:math.GR/0102078).
  • 7. Peter Doubilet, Gian-Carlo Rota, and Richard Stanley, On the foundations of combinatorial theory. VI. The idea of generating function, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 267–318. MR 0403987
  • 8. Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
  • 9. James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460
  • 10. Bernadette Krawczyk and Roland Speicher, Combinatorics of free cumulants, J. Combin. Theory Ser. A 90 (2000), no. 2, 267–292. MR 1757277, 10.1006/jcta.1999.3032
  • 11. G. Kreweras, Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), no. 4, 333–350 (French). MR 0309747
  • 12. Alexandru Nica and Roland Speicher, A “Fourier transform” for multiplicative functions on non-crossing partitions, J. Algebraic Combin. 6 (1997), no. 2, 141–160. MR 1436532, 10.1023/A:1008643104945
  • 13. Alexandru Nica and Roland Speicher, On the multiplication of free 𝑁-tuples of noncommutative random variables, Amer. J. Math. 118 (1996), no. 4, 799–837. MR 1400060
  • 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. Victor Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177 (1997), no. 1-3, 195–222. MR 1483446, 10.1016/S0012-365X(96)00365-2
  • 17. Roland Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 298 (1994), no. 4, 611–628. MR 1268597, 10.1007/BF01459754
  • 18. Roland Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627, x+88. MR 1407898, 10.1090/memo/0627
  • 19. Dan Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), no. 3, 323–346. MR 839105, 10.1016/0022-1236(86)90062-5
  • 20. D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L54, 05E15, 20F55

Retrieve articles in all journals with MSC (2000): 46L54, 05E15, 20F55

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