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Commuting Families in Hecke and Temperley-Lieb Algebras

Published online by Cambridge University Press:  11 January 2016

Tom Halverson ,
Manuela Mazzocco  and
Arun Ram
Tom Halverson
Affiliation:
Department of MathematicsMacalester College, Saint Paul, MN 55105, U.S.A.halverson@macalester.edu
Manuela Mazzocco
Affiliation:
Department of MathematicsUniversity of Wisconsin, Madison, WI 53706, U.S.A.
Arun Ram
Affiliation:
Department of Mathematics and StatisticsUniversity of Melbourne, Parkville VIC 3010, Australiaaram@unimelb.edu.au, Department of MathematicsUniversity of Wisconsin Madison, WI 53706, U.S.A.
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Abstract

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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 . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

Keywords

20G0516G9981R5082B20
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 195 , 2009 , pp. 125 - 152
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

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