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

Let (W, S) be an affine Coxeter system of type B̃ or D̃ and 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 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 . Introduction The Temperley-Lieb (TL) algebra is a very classical mathematical object studied in algebra, combinatorics, statistical mechanics and mathematical physics, introduced by Temperley and Lieb in 1971 [16]. Kauffman [13] and Penrose [14] showed that the TL algebra can be realized as a diagram algebra, that is an associative algebra with a basis given by certain diagrams on the plane. On the other hand, Jones presented the TL algebra in terms of abstract generators and relations. In [12], he also showed that this algebra occurs naturally as a quotient of the Hecke algebra of type A. The realization of the TL algebra as a Hecke algebra quotient was generalized by Graham in [6]. He defined the so-called generalized Temperley-Lieb algebra TL(Γ) for a Coxeter system of arbitrary type Γ and showed that it has a monomial basis indexed by the fully commutative elements of the underlying Coxeter group. During the years, diagrammatic representations for TL(Γ) have been found for each Coxeter system of finite type and for two systems of affine type. More precisely, in [8], [10] and [7] Green defined a diagram calculus in finite Coxeter types B, D, E and H. For affine types, in [5] Fan and Green provided a realization of TL(Ã) as a diagram algebra on a cylinder and, more recently, in [3, 4], Ernst represented TL(C̃) as an algebra of decorated diagrams. The aim of this paper is to give diagrammatic representations for the two remaining affine families, namely types B̃ and D̃. Taking inspiration from Ernst work on type C̃, we consider two subsets of the decorated diagrams introduced in [3], we endow them with an algebra structure, and then we define, modulo some relations, two new quotient algebras, denoted by D(B̃n+1) and D(D̃n+2). Our main results show that these algebras are actually isomorphic to TL(B̃) and TL(D̃), respectively. In order to do so, we first give an explicit description of two special families of irreducible decorated diagrams, called admissible, that turn out to be linear bases of D(B̃n+1) and D(D̃n+2), respectively. We then introduce a notion of length for admissible diagrams that measures Date: December 23, 2022.