Towards a classification of multi-faced independences: a combinatorial approach

Gerhold, Malte; Varšo, Philipp

doi:10.5802/alco.356

Gerhold, Malte ¹; Varšo, Philipp

¹ Saarland University Faculty of Mathematics Postbox 151150 D-66041 Saarbruecken Germany

Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 679-711.

Abstract

We determine a set of necessary conditions on a partition-indexed family of complex numbers to be the “highest coefficients” of a positive and symmetric multi-faced universal product, i.e. the product associated with a multi-faced version of noncommutative stochastic independence, such as bifreeness. The highest coefficients of a universal product are the weights of the moment-cumulant relation for its associated independence. We show that these conditions are almost sufficient, in the sense that whenever the conditions are satisfied, one can associate a (automatically unique) symmetric universal product with the prescribed highest coefficients. Furthermore, we give a quite explicit description of such families of coefficients, thereby producing a list of candidates that must contain all positive symmetric universal products. We discover in this way four (three up to trivial face-swapping) previously unknown moment-cumulant relations that give rise to symmetric universal products; to decide whether they are positive, and thus give rise to independences which can be used in an operator algebraic framework, remains an open problem.

Received: 2023-02-03
Revised: 2023-05-03
Accepted: 2024-01-16
Published online: 2024-06-27

DOI: 10.5802/alco.356

Classification: 46L53, 05A18, 60A05, 18M05, 46L54
Keywords: noncommutative probability, multi-faced independence, cumulants, set partitions

Author's affiliations:

Gerhold, Malte ¹; Varšo, Philipp

¹ Saarland University Faculty of Mathematics Postbox 151150 D-66041 Saarbruecken Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Gerhold, Malte; Varšo, Philipp. Towards a classification of multi-faced independences: a combinatorial approach. Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 679-711. doi : 10.5802/alco.356. https://alco.centre-mersenne.org/articles/10.5802/alco.356/

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