Non-crossing cumulants of type B

Biane, Philippe; Goodman, Frederick M.; Nica, Alexandru

doi:10.1090/s0002-9947-03-03196-9

Cited by 43 publications

(83 citation statements)

References 16 publications

(35 reference statements)

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“…(a) This framework generalizes the link-algebra associated to a noncommutative probability space of type B, in the sense introduced by Biane, Goodman and Nica [2]. One can thus take the point of view that (1.1) provides us with an enlarged framework for doing "free probability of type B".…”

Section: The Framework Of the Papermentioning

confidence: 90%

Infinitesimal non-crossing cumulants and free probability of type B

Fevrier¹,

Nica²

2010

Journal of Functional Analysis

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Free probabilistic considerations of type B first appeared in the paper of Biane, Goodman and Nica [P. Biane, F. Goodman, A. Nica, Non-crossing cumulants of type B, Trans. Amer. Math. Soc. 355 (2003Soc. 355 ( ) 2263Soc. 355 ( -2303. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko [S.T. Belinschi, D. Shlyakhtenko, Free probability of type B: Analytic aspects and applications, preprint, 2009, available online at www.arxiv.org under reference arXiv:0903.2721]. The interplay between "type B" and "infinitesimal" is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A 1 , . . . , A k in an infinitesimal noncommutative probability space (A, ϕ, ϕ ) and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A, ϕ, ϕ ), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A 1 , . . . , A k is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal counterpart for a theorem of Speicher from "usual" free probability. We show that the lattices NC (B) (n) of non-crossing partitions of type B appear in the combinatorial study of (A, ϕ, ϕ ), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A 1 , . . . , A k in (A, ϕ)

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Section: The Framework Of the Papermentioning

confidence: 90%

Infinitesimal non-crossing cumulants and free probability of type B

Fevrier¹,

Nica²

2010

Journal of Functional Analysis

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“…Coxeter-plane diagrams of parabolic subgroups are centrally symmetric set partitions of the vertices of the 2n-gon. Noncrossing Criterion 1 is the combinatorial crossing/noncrossing criterion of [19], which has been shown in [4,6,10] to correctly classify parabolic subgroups as crossing/noncrossing in the algebraic sense. Again, to correctly compare the above construction to [4,6,10,19], one must take the correct 2n-cycle c described above.…”

Section: 1mentioning

confidence: 99%

Noncrossing partitions, clusters and the Coxeter plane

Reading¹

2009

Preprint

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When W is a finite Coxeter group of classical type (A, B, or D), noncrossing partitions associated to W and compatibility of almost positive roots in the associated root system are known to be modeled by certain planar diagrams. We show how the classical-type constructions of planar diagrams arise uniformly from projections of small W -orbits to the Coxeter plane. When the construction is applied beyond the classical cases, simple criteria are apparent for noncrossing and for compatibility for W of types H 3 and I 2 (m) and less simple criteria can be found for compatibility in types E 6 , F 4 and H 4 . Our construction also explains why simple combinatorial models are elusive in the larger exceptional types.

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“…Now many extensions and generalizations of free probability have been discovered. Infinitesimal freeness is one of the generalization of freeness [4], [2]. An application of infinitesimal freeness to random matrix theory was first presented by Shlyakhtenko [20], and he showed how to understand the finite rank perturbation in some random matrix models by applying infinitesimal freenss.…”

Section: Introductionmentioning

confidence: 99%

Operator-Valued Infinitesimal Multiplicative Convolutions

Tseng¹

2022

Preprint

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We consider the notions of operator-valued infinitesimal (OVI) free independence, OVI Boolean independence, and OVI monotone independence. For each notion of OVI independence, we introduce the corresponding infinitesimal transforms, and then we show that the transforms satisfy certain multiplicative property. We also provide an application on complex Wishart matrices via our infinitesimal free multiplicative formula.

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Non-crossing cumulants of type B

Cited by 43 publications

References 16 publications

Infinitesimal non-crossing cumulants and free probability of type B

Infinitesimal non-crossing cumulants and free probability of type B

Noncrossing partitions, clusters and the Coxeter plane

Operator-Valued Infinitesimal Multiplicative Convolutions

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