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

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 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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Submitted on : Monday, July 6, 2020 - 10:28:35 AM
Last modification on : Wednesday, September 14, 2022 - 5:42:26 PM

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Nicolas Crampé, Loïc Poulain d'Andecy, Luc Vinet. Temperley–Lieb, Brauer and Racah algebras and other centralizers of $\mathfrak {su}(2)$. Transactions of the American Mathematical Society, 2020, 373 (7), pp.4907-4932. ⟨10.1090/tran/8055⟩. ⟨hal-02890155⟩



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