Nagoya Mathematical Journal

Commuting families in Hecke and Temperley-Lieb algebras

Tom Halverson, Manuela Mazzocco, and Arun Ram

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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.

Article information

Source
Nagoya Math. J., Volume 195 (2009), 125-152.

Dates
First available in Project Euclid: 14 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1252934375

Mathematical Reviews number (MathSciNet)
MR2552957

Zentralblatt MATH identifier
1217.20002

Subjects
Primary: 20G05: Representation theory
Secondary: 16G99: None of the above, but in this section 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Citation

Halverson, Tom; Mazzocco, Manuela; Ram, Arun. Commuting families in Hecke and Temperley-Lieb algebras. Nagoya Math. J. 195 (2009), 125--152. https://projecteuclid.org/euclid.nmj/1252934375


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