2002/06/17 by Philippe Biane, Frederick M. Goodman, Frederick Goodman +4
Mathematics · #05A18 #46L54 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Operator Algebras (math.OA) #Random Matrices and Applications #math.CO #math.OA #msc:05A18 #msc:46L54
paper · pdf · doi:10.48550/arxiv.math/0206167
48 pages, latex
arxiv created 2002/06/17 · openalex publication_date 2002/06/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 \em 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.