A “Fourier Transform” for Multiplicative Functions on Non-Crossing Partitions
Alexandru Nica
and Roland Speicher
DOI: 10.1023/A:1008643104945
Abstract
We describe the structure of the group of normalized multiplicative functions on lattices of non-crossing partitions. As an application, we give a combinatorial proof of a theorem of D. Voiculescu concerning the multiplication of free random variables
Pages: 141–160
Keywords: non-crossing partition; moebius function; free random variables
Full Text: PDF
References
1. Ph. Biane, “Some properties of crossings and partitions,” preprint.
2. Ph. Biane, “Minimal factorizations of a cycle and central multiplicative functions on the infinite symmetric group,” preprint.
3. 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. (Ed.), University of California Press, 1972, pp. 267-318.
4. I.P. Goulden and D.M. Jackson, “Symmetric functions and Macdonald's result for top connexion coefficients in the symmetric group,” Journal of Algebra 166 (1994), 364-378.
5. U. Haagerup, “On Voiculescu's R- and S-transforms for free non-commuting random variables,” preprint.
6. D.M. Jackson, “Some combinatorial problems associated with products of conjugacy classes of the symmetric group,” J. Comb. Theory 49 (1988), 363-369.
7. G. Kreweras, “Sur les partitions non-croisées d'un cycle,” Discrete Math. 1 (1972), 333-350.
8. M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and its Applications, Addison-Wesley, 1983, vol. 17.
9. A. Nica and R. Speicher, “On the multiplication of free n-tuples of non-commutative random variables,” With an Appendage 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.
10. G.-C. Rota, “On the foundations of combinatorial theory (I): Theory of Moebius functions,” Z. Wahrscheinlichkeitstheorie verw. Geb. 2 (1964), 340-368.
11. G.-C. Rota, Finite Operator Calculus, Academic Press, 1975.
12. R. Simion and D. Ullman, “On the structure of the lattice of non-crossing partitions,” Discrete Math. 98 (1991), 193-206.
13. R. Speicher, “Multiplicative functions on the lattice of non-crossing partitions and free convolution,” Math. Annalen 298 (1994), 611-628.
14. R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Habilitationsschrift, Heidelberg, 1994.
15. R.P. Stanley, Enumerative Combinatorics, Wadsworth & Brooks / Cole Mathematics Series, 1986, vol. I.
16. D. Voiculescu, “Symmetries of some reduced free product C*-algebras,” in Operator Algebras and Their Connection with Topology and Ergodic Theory (Springer Lecture Notes in Mathematics) 1132 (1985), 556- 588.
17. D. Voiculescu, “Addition of certain non-commuting random variables,” J. Funct. Anal. 66 (1986), 323-346.
18. D. Voiculescu, “Multiplication of certain non-commuting random variables,” J. Operator Theory 18 (1987), 223-235.
19. D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series (published by the AMS), 1993, vol. 1.
2. Ph. Biane, “Minimal factorizations of a cycle and central multiplicative functions on the infinite symmetric group,” preprint.
3. 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. (Ed.), University of California Press, 1972, pp. 267-318.
4. I.P. Goulden and D.M. Jackson, “Symmetric functions and Macdonald's result for top connexion coefficients in the symmetric group,” Journal of Algebra 166 (1994), 364-378.
5. U. Haagerup, “On Voiculescu's R- and S-transforms for free non-commuting random variables,” preprint.
6. D.M. Jackson, “Some combinatorial problems associated with products of conjugacy classes of the symmetric group,” J. Comb. Theory 49 (1988), 363-369.
7. G. Kreweras, “Sur les partitions non-croisées d'un cycle,” Discrete Math. 1 (1972), 333-350.
8. M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and its Applications, Addison-Wesley, 1983, vol. 17.
9. A. Nica and R. Speicher, “On the multiplication of free n-tuples of non-commutative random variables,” With an Appendage 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.
10. G.-C. Rota, “On the foundations of combinatorial theory (I): Theory of Moebius functions,” Z. Wahrscheinlichkeitstheorie verw. Geb. 2 (1964), 340-368.
11. G.-C. Rota, Finite Operator Calculus, Academic Press, 1975.
12. R. Simion and D. Ullman, “On the structure of the lattice of non-crossing partitions,” Discrete Math. 98 (1991), 193-206.
13. R. Speicher, “Multiplicative functions on the lattice of non-crossing partitions and free convolution,” Math. Annalen 298 (1994), 611-628.
14. R. Speicher, Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory, Habilitationsschrift, Heidelberg, 1994.
15. R.P. Stanley, Enumerative Combinatorics, Wadsworth & Brooks / Cole Mathematics Series, 1986, vol. I.
16. D. Voiculescu, “Symmetries of some reduced free product C*-algebras,” in Operator Algebras and Their Connection with Topology and Ergodic Theory (Springer Lecture Notes in Mathematics) 1132 (1985), 556- 588.
17. D. Voiculescu, “Addition of certain non-commuting random variables,” J. Funct. Anal. 66 (1986), 323-346.
18. D. Voiculescu, “Multiplication of certain non-commuting random variables,” J. Operator Theory 18 (1987), 223-235.
19. D.V. Voiculescu, K.J. Dykema, and A. Nica, Free Random Variables, CRM Monograph Series (published by the AMS), 1993, vol. 1.