## Commuting families in Hecke and Temperley-Lieb Algebras

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

Last update: 29 January 2014

## 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 ${U}_{h}{\U0001d524\U0001d529}_{n}\text{.}$
We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

## Notes and references

This is a typed version of *Commuting families in Hecke and Temperley-Lieb Algebras* by Tom Halverson, Manuela Mazzocco and Arun Ram.

AMS Subject Classifications: Primary 20G05; Secondary 16G99, 81R50, 82B20.

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