Non-crossing linked partitions, the partial order ≪ on 𝑁𝐶(𝑛), and the 𝑆-transform

Nica, Alexandru

doi:10.1090/S0002-9939-09-10218-6

Non-crossing linked partitions, the partial order $\ll$ on $NC(n)$, and the $S$-transform
HTML articles powered by AMS MathViewer

by Alexandru Nica PDF

Proc. Amer. Math. Soc. 138 (2010), 1273-1285 Request permission

Abstract:

The paper establishes a connection between two recent combinatorial developments in free probability: the non-crossing linked partitions introduced by Dykema in 2007 to study the $S$-transform, and the partial order $\ll$ on $NC(n)$ introduced by Belinschi and Nica in 2008 in order to study relations between free and Boolean probability. More precisely, one has a canonical bijection between $NCL(n)$ (the set of all non-crossing linked partitions of $\{ 1, \ldots , n \}$) and the set $\{ ( \alpha , \beta ) \mid \alpha , \beta \in NC(n), \ \alpha \ll \beta \}$. As a consequence of this bijection, one gets an alternative description of Dykema’s formula expressing the moments of a non-commutative random variable $a$ in terms of the coefficients of the reciprocal $S$-transform $1/S_a$. Moreover, due to the Boolean features of $\ll$, this formula can be simplified to a form which resembles the moment-cumulant formula from $c$-free probability.

References

Serban T. Belinschi and Alexandru Nica, $\eta$-series and a Boolean Bercovici-Pata bijection for bounded $k$-tuples, Adv. Math. 217 (2008), no. 1, 1–41. MR 2357321, DOI 10.1016/j.aim.2007.06.015
Serban T. Belinschi and Alexandru Nica, Free Brownian motion and evolution towards $\boxplus$-infinite divisibility for $k$-tuples, Internat. J. Math. 20 (2009), no. 3, 309–338. MR 2500073, DOI 10.1142/S0129167X09005303
Marek Bożejko, Michael Leinert, and Roland Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996), no. 2, 357–388. MR 1432836, DOI 10.2140/pjm.1996.175.357
William Y. C. Chen, Susan Y. J. Wu, and Catherine H. Yan, Linked partitions and linked cycles, European J. Combin. 29 (2008), no. 6, 1408–1426. MR 2423730, DOI 10.1016/j.ejc.2007.06.022
Kenneth J. Dykema, Multilinear function series and transforms in free probability theory, Adv. Math. 208 (2007), no. 1, 351–407. MR 2304321, DOI 10.1016/j.aim.2006.02.011
M. Mastnak, A. Nica. Hopf algebras and the logarithm of the $S$-transform, to appear in Transactions of the American Mathematical Society, arXiv:0807.4169.
Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR 2266879, DOI 10.1017/CBO9780511735127
M. Popa. Non-crossing linked partitions and multiplication of free random variables, preprint, 2008, arXiv:0812.2064.
Rodica Simion, Noncrossing partitions, Discrete Math. 217 (2000), no. 1-3, 367–409 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766277, DOI 10.1016/S0012-365X(99)00273-3
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
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
Dan Voiculescu, Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), no. 2, 223–235. MR 915507
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