### Commuting families in Hecke and Temperley-Lieb algebras

Tom Halverson, Manuela Mazzocco, and Arun Ram
Source: Nagoya Math. J. Volume 195 (2009), 125-152.

#### Abstract

We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group $U_{h}\mathfrak{gl}_{n}$. We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

First Page:
Primary Subjects: 20G05
Secondary Subjects: 16G99, 81R50, 82B20
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.nmj/1252934375
Zentralblatt MATH identifier: 05611433
Mathematical Reviews number (MathSciNet): MR2552957

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