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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Temperley-Lieb, Brauer and Racah algebras and other centralizers of $ \mathfrak{su}(2)$


Authors: Nicolas Crampé, Loïc Poulain d’Andecy and Luc Vinet
Journal: Trans. Amer. Math. Soc. 373 (2020), 4907-4932
MSC (2010): Primary 16S20, 17B35
DOI: https://doi.org/10.1090/tran/8055
Published electronically: March 27, 2020
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Abstract: In the spirit of the Schur-Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of $ \mathfrak{su}(2)$. As a first step we show that the Racah algebra always surjects onto the centralizer. We then offer a conjecture regarding the description of the kernel of the map, which depends on the irreducible representations. If true, this conjecture would provide a presentation of the centralizer as a quotient of the Racah algebra. We prove this conjecture in several cases. In particular, while doing so, we explicitly obtain the Temperley-Lieb algebra, the Brauer algebra and the one-boundary Temperley-Lieb algebra as quotients of the Racah algebra.


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Additional Information

Nicolas Crampé
Affiliation: Institut Denis-Poisson CNRS/UMR 7013, Université de Tours, Université d’Orléans, Parc de Grandmont, 37200 Tours, France; and Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec, H3C 3J7, Canada
Email: crampe1977@gmail.com

Loïc Poulain d’Andecy
Affiliation: Laboratoire de mathématiques de Reims UMR 9008, Université de Reims Champagne-Ardenne, UFR Sciences exactes et naturelles, Moulin de la Housse BP 1039, 51100 Reims, France
Email: loic.poulain-dandecy@univ-reims.fr

Luc Vinet
Affiliation: Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec, H3C 3J7, Canada
Email: vinet@CRM.UMontreal.ca

DOI: https://doi.org/10.1090/tran/8055
Received by editor(s): August 2, 2019
Received by editor(s) in revised form: November 4, 2019
Published electronically: March 27, 2020
Additional Notes: The first and second authors were partially supported by Agence National de la Recherche Projet AHA ANR-18-CE40-0001.
The research of the third author was supported in part by a Discovery Grant from the Natural Science and Engineering Research Council (NSERC) of Canada.
Article copyright: © Copyright 2020 American Mathematical Society