Non-crossing cumulants of type B |
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Authors: | Philippe Biane Frederick Goodman Alexandru Nica |
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Institution: | Département de Mathématiques et Applications de l'Ecole Normale Supérieure, 45 rue d'Ulm, 75230 Paris, France ; Department of Mathematics, University of Iowa, Iowa City, Iowa 52242 ; Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada |
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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. |
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