Diagram Calculus for the Affine Temperley--Lieb Algebra of Type $D$

TL;DR

This paper introduces a diagrammatic calculus for the affine Temperley-Lieb algebra of type D, providing an explicit diagram basis that corresponds bijectively to the algebra's classical basis, thus offering a new combinatorial perspective.

Contribution

It defines an infinite dimensional diagram algebra for the affine type D Temperley-Lieb algebra and establishes a bijective correspondence with its classical basis, enhancing combinatorial understanding.

Findings

Established an isomorphism between the diagram algebra and TL(W).

Provided an explicit diagram basis indexed by fully commutative elements.

Connected diagrammatic and algebraic bases through bijective correspondence.

Abstract

Let (W,S) be a Coxeter system of affine type D, and let TL(W) the corresponding generalized Temperley-Lieb algebra. In this extended abstract we define an infinite dimensional associative algebra made of decorated diagrams which is isomorphic to TL(W). Moreover, we describe an explicit basis for such an algebra of diagrams which is in bijective correspondence with the classical monomial basis of TL(W), indexed by the fully commutative elements of W.