Diagrammatic Representations of Generalized Temperley–Lieb Algebras of Affine Types $\widetilde{D}$ and $\widetilde{B}$

Let $(W,S)$ be an affine Coxeter system of type $\Gamma$, with $\Gamma$ equal to $\widetilde{D}$ or $\widetilde{B}$, and $\operatorname{TL}(\Gamma)$ the corresponding generalized Temperley–Lieb algebra. In this paper we define an infinite dimensional associative algebra made of decorated diagrams that is isomorphic to $\operatorname{TL}(\Gamma)$. 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 $W$.