Cyclotomic Temperley–Lieb Algebra of Type D and Its Representation Theory

Abstract

We define a new class of algebras, cyclotomic Temperley–Lieb algebras of type D, in a diagrammatic way, which is a generalization of Temperley–Lieb algebras of type D. We prove that the cyclotomic Temperley–Lieb algebras of type D are cellular. In fact, an explicit cellular basis is given by means of combinatorial methods. After determining all the irreducible representations of these algebras, we give a necessary and sufficient condition for a cyclotomic Temperley–Lieb algebra of type D to be quasi-hereditary.