Non-crossing cumulants of type B

Biane, Philippe; Goodman, Frederick; Nica, Alexandru

doi:10.1090/S0002-9947-03-03196-9

Non-crossing cumulants of type B
HTML articles powered by AMS MathViewer

by Philippe Biane, Frederick Goodman and Alexandru Nica PDF

Trans. Amer. Math. Soc. 355 (2003), 2263-2303 Request permission

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

D. Bessis. The dual braid monoid, preprint, January 2002 (arXiv:math.GR/0101158).
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, DOI 10.1006/jcta.1996.0101
Philippe Biane, Some properties of crossings and partitions, Discrete Math. 175 (1997), no. 1-3, 41–53. MR 1475837, DOI 10.1016/S0012-365X(96)00139-2
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.
Thomas Brady, A partial order on the symmetric group and new $K(\pi ,1)$’s for the braid groups, Adv. Math. 161 (2001), no. 1, 20–40. MR 1857934, DOI 10.1006/aima.2001.1986
T. Brady and C. Watt. $K( \pi , 1)$’s for Artin groups of finite type, preprint, February 2001 (arXiv:math.GR/0102078).
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
Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Bernadette Krawczyk and Roland Speicher, Combinatorics of free cumulants, J. Combin. Theory Ser. A 90 (2000), no. 2, 267–292. MR 1757277, DOI 10.1006/jcta.1999.3032
G. Kreweras, Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), no. 4, 333–350 (French). MR 309747, DOI 10.1016/0012-365X(72)90041-6
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, DOI 10.1023/A:1008643104945
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, DOI 10.1353/ajm.1996.0034
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.
A. Nica, D. Shlyakhtenko, and R. Speicher. Operator-valued distributions I. Characterizations of freeness, International Mathematics Research Notices 29 (2002), 1509-1538.
Victor Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177 (1997), no. 1-3, 195–222. MR 1483446, DOI 10.1016/S0012-365X(96)00365-2
Roland Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 298 (1994), no. 4, 611–628. MR 1268597, DOI 10.1007/BF01459754
Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
Dan Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), no. 3, 323–346. MR 839105, DOI 10.1016/0022-1236(86)90062-5
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, DOI 10.1090/crmm/001