COMMUTING FAMILIES IN HECKE AND TEMPERLEY-LIEB ALGEBRAS

Abstract

<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0027763000009740_inline1" />. We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.</jats:p>