Commuting Families in Temperley-Lieb Algebras

Details Commuting Families in Temperley-Lieb Algebras Commuting Families in Temperley-Lieb Algebras

2007-10-02

arxiv.org/abs/0710.0596v1

Mathematics

Representation Theory

20 pages

Scientific paper

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.

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