The representation theory of cyclotomic Temperley-Lieb algebras

Abstract

A class of associative algebras called cyclotomic Temperley-Lieb algebras is introduced in terms of generators and relations. They are closely related to the group algebras of complex reflection groups on the one hand and generalizations of the usual Temperley-Lieb algebras on the other hand. It is shown that the cyclotomic Temperley-Lieb algebras can be defined by means of labelled Temperley-Lieb diagrams and are cellular in the sense of Graham and Lehrer. One thus obtains not only a description of the irreducible representations, but also a criterion for their quasi-heredity in the sense of Cline, Parshall and Scott. The branching rule for cell modules and the determinants of Gram matrices for certain cell modules are calculated, they can be expressed in terms of generalized Tchebychev polynomials, which therefore play an important role for semisimplicity.