Cyclotomic Temperley-Lieb algebra of type D and its representation theory

TL;DR

This paper introduces cyclotomic Temperley-Lieb algebras of type D, proves their cellularity, and characterizes their irreducible representations and quasi-hereditary conditions, advancing the understanding of their algebraic structure.

Contribution

It defines a new class of algebras, establishes their cellular structure, and fully determines their irreducible representations and quasi-hereditary criteria.

Findings

Algebras are cellular with explicit bases.

All irreducible representations are classified.

Conditions for quasi-heredity are provided.

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.