Diagrammatic representations of Generalized Temperley-Lieb algebras of affine type $\widetilde{B}$ and $\widetilde{D}$

TL;DR

This paper constructs a diagrammatic algebra for affine type B and D Temperley-Lieb algebras, providing an explicit basis linked to classical monomial bases and fully commutative elements.

Contribution

It introduces a diagrammatic algebra isomorphic to the generalized Temperley-Lieb algebra of affine types B and D, with an explicit admissible diagram basis.

Findings

Defined an infinite dimensional diagrammatic algebra for affine types B and D

Established an explicit basis of admissible decorated diagrams

Proved bijection between basis diagrams and classical monomial basis

Abstract

Let $(W,S)$ be an affine Coxeter system of type $\widetilde{B}$ or $\widetilde{D}$ and ${\rm TL}(W)$ the corresponding generalized Temperley-Lieb algebra. In this paper we define an infinite dimensional associative algebra made of decorated diagrams that is isomorphic to ${\rm TL}(W)$. Moreover, we describe an explicit basis for such an algebra consisting of special decorated diagrams that we call admissible. Such basis is in bijective correspondence with the classical monomial basis of the generalized Temperley-Lieb algebra indexed by the fully commutative elements of $W$.