Hopf algebras and the logarithm of the $S$-transform in free probability
HTML articles powered by AMS MathViewer
- by Mitja Mastnak and Alexandru Nica PDF
- Trans. Amer. Math. Soc. 362 (2010), 3705-3743 Request permission
Abstract:
Let $k$ be a positive integer and let $\mathcal {G}_k$ denote the set of all joint distributions of $k$-tuples $(a_1,\ldots ,a_k)$ in a noncommutative probability space $(\mathcal {A},\varphi )$ such that $\varphi (a_1)=\cdots =\varphi (a_k) = 1$. $\mathcal {G}_k$ is a group under the operation of the free multiplicative convolution $\boxtimes$. We identify $\bigl ( \mathcal {G}_k, \boxtimes \bigr )$ as the group of characters of a certain Hopf algebra $\mathcal {Y}^{(k)}$. Then, by using the log map from characters to infinitesimal characters of $\mathcal {Y}^{(k)}$, we introduce a transform $LS_{\mu }$ for distributions $\mu \in \mathcal {G}_k$. $LS_{\mu }$ is a power series in $k$ noncommuting indeterminates $z_1, \ldots , z_k$; its coefficients can be computed from the coefficients of the $R$-transform of $\mu$ by using summations over chains in the lattices $NC(n)$ of noncrossing partitions. The $LS$-transform has the “linearizing” property that \[ LS_{\mu \boxtimes \nu } =LS_{\mu } +LS_{\nu }, \ \ \forall \mu , \nu \in \mathcal {G}_k \mbox { such that } \mu \boxtimes \nu = \nu \boxtimes \mu . \]
In the particular case $k=1$ one has that ${\mathcal Y}^{(1)}$ is naturally isomorphic to the Hopf algebra $\mbox {Sym}$ of symmetric functions and that the $LS$-transform is very closely related to the logarithm of the $S$-transform of Voiculescu by the formula \[ LS_{\mu } (z) = -z \log S_{\mu } (z), \ \ \forall \mu \in \mathcal {G}_1. \] In this case the group $(\mathcal G_1, \boxtimes )$ can be identified as the group of characters of $\mbox {Sym}$, in such a way that the $S$-transform, its reciprocal $1/S$ and its logarithm $\log S$ relate in a natural sense to the sequences of complete, elementary and, respectively, power sum symmetric functions.
References
- Marcelo Aguiar, Nantel Bergeron, and Frank Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math. 142 (2006), no. 1, 1–30. MR 2196760, DOI 10.1112/S0010437X0500165X
- Michael Anshelevich, Edward G. Effros, and Mihai Popa, Zimmermann type cancellation in the free Faà di Bruno algebra, J. Funct. Anal. 237 (2006), no. 1, 76–104. MR 2239259, DOI 10.1016/j.jfa.2005.12.009
- 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
- Richard Ehrenborg, On posets and Hopf algebras, Adv. Math. 119 (1996), no. 1, 1–25. MR 1383883, DOI 10.1006/aima.1996.0026
- HĂ©ctor Figueroa and JosĂ© M. Gracia-BondĂa, Combinatorial Hopf algebras in quantum field theory. I, Rev. Math. Phys. 17 (2005), no. 8, 881–976. MR 2167639, DOI 10.1142/S0129055X05002467
- Uffe Haagerup, On Voiculescu’s $R$- and $S$-transforms for free non-commuting random variables, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 127–148. MR 1426838, DOI 10.1215/s0012-7094-97-09004-9
- S. A. Joni and G.-C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979), no. 2, 93–139. MR 544721, DOI 10.1002/sapm197961293
- 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
- I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
- 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, 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
- William R. Schmitt, Incidence Hopf algebras, J. Pure Appl. Algebra 96 (1994), no. 3, 299–330. MR 1303288, DOI 10.1016/0022-4049(94)90105-8
- 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
- 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
- Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
- 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
- Andrey V. Zelevinsky, Representations of finite classical groups, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin-New York, 1981. A Hopf algebra approach. MR 643482
Additional Information
- Mitja Mastnak
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- Address at time of publication: Department of Mathematics and Computer Science, Saint Mary’s University, Halifax, Nova Scotia, Canada B3H 3C3
- MR Author ID: 695207
- Email: mmastnak@cs.smu.ca
- Alexandru Nica
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- Email: anica@math.uwaterloo.ca
- Received by editor(s): August 12, 2008
- Published electronically: February 8, 2010
- Additional Notes: The research of the second-named author was supported by a Discovery Grant from NSERC, Canada
- © Copyright 2010 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 3705-3743
- MSC (2010): Primary 46L54; Secondary 16T30
- DOI: https://doi.org/10.1090/S0002-9947-10-04995-0
- MathSciNet review: 2601606