De Finetti Theorems for the Unitary Dual Group

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• ISSN: 1815-0659
• Main Authors: Baraquin, Isabelle, Cébron, Guillaume, Franz, Uwe, Maassen, Laura, Weber, Moritz
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211721
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: De Finetti Theorems for the Unitary Dual Group. Isabelle Baraquin, Guillaume Cébron, Uwe Franz, Laura Maassen and Moritz Weber. SIGMA 18 (2022), 067, 29 pages

We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing R-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in *-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group ⁺ₙ. Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in *-probability spaces. On the other hand, if we drop the assumption of faithful states in *-probability spaces, we obtain a non-trivial half of a de Finetti theorem similar to the case of the dual group action.