Temperley–Lieb, Brauer and Racah algebras and other centralizers of $\mathfrak {su}(2)$
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- by Nicolas Crampé, Loïc Poulain d’Andecy and Luc Vinet PDF
- Trans. Amer. Math. Soc. 373 (2020), 4907-4932 Request permission
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.References
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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
- MR Author ID: 178665
- ORCID: 0000-0001-6211-7907
- Email: vinet@CRM.UMontreal.ca
- 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. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 4907-4932
- MSC (2010): Primary 16S20, 17B35
- DOI: https://doi.org/10.1090/tran/8055
- MathSciNet review: 4127866