Abstract. We establish the existence of an IC basis for the generalized Temperley–Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is finite dimensional. To appear in Mathematical Research Letters

We study a finite-dimensional quotient of the Hecke algebra of type Hn for general n, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring.

The recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor. The method is hence adapted to treat sequences that are not necessarily towers by inclusion, such as symplectic blob algebras (diagram algebra quotients of the type- ˜ C Hecke algebras). By carefully reviewing the diagram algebra construction, we find a new set of functors interrelating module categories of ordinary blob algebras (diagram algebra quotients of the type-B Hecke algebras) at different values of the algebra parameters. We show that these functors generalise to determine the structure of symplectic blob algebras, and hence of certain two-boundary Temperley-Lieb algebras arising in Statistical Mechanics. We identify the diagram basis with a cellular basis for each symplectic blob algebra, and prove that these algebras are quasihereditary over a field for almost all parameter choices, and generically semisimple. (That is, we give

. We investigate the compatibility of the set of fully commutative elements of a Coxeter group with the various types of Kazhdan{Lusztig cells using a canonical basis for a generalized version of the Temperley{Lieb algebra. Cellules pleinement commutatives de Kazhdan{Lusztig Nous etudions la compatibilite entre l'ensemble des elements pleinement commutatifs d'un groupe de Coxeter et les divers types de cellules de Kazhdan{Lusztig, en utilisant une base canonique pour une version generalisee de l'algebre de Temperley{ Lieb. Key Words: canonical basis, cell theory, Coxeter group, Hecke algebra, Kazhdan{ Lusztig basis, Temperley{Lieb algebra The rst author was supported in part by a NUF{NAL award from the Nueld Foundation. Typeset by A M S-T E X 1 2 R.M. GREEN AND J. LOSONCZY Introduction The fully commutative elements, W c , of a Coxeter group W may be dened, following [17], as the set of elements w with the property that any reduced expression for w may be obtai...

: Let H be a Hecke algebra associated with a Coxeter system of type D, and let TL be the corresponding Temperley{Lieb quotient. The algebra TL admits a canonical basis, which facilitates the construction of irreducible representations. In this paper, we explain the relationship between the canonical basis of TL and the Kazhdan{Lusztig basis of H. Key Words: canonical basis; Coxeter group; Hecke algebra; Kazhdan{Lusztig basis; Temperley{Lieb algebra. 2 1. Introduction Let X be a Coxeter graph and let W (X) be an associated Coxeter group with Coxeter generators S(X) and length function `. Let H(X) be the corresponding Hecke algebra. This is an associative, unital algebra over the ring A = Z[v; v 1 ] of Laurent polynomials. The Hecke algebra H(X) has generators T s , one for each s 2 S(X), which are subject to the following relations: T 2 s = (q 1)T s + q, where q = v 2 ; (T s T s 0 ) m = (T s 0 T s ) m if ss 0 has order 2m; and (T s T s 0 ) m T s = (T s 0 T s ) m...

. We compare the canonical basis for a generalized Temperley{Lieb algebra of type A or B with the Kazhdan{Lusztig basis for the corresponding Hecke algebra. Introduction Generalized Temperley{Lieb algebras arise as certain quotients of Hecke algebras associated to Coxeter systems in the same way that the ordinary Temperley{Lieb algebra can be realised as a quotient of the Hecke algebra of type A (see [9]). The nite-dimensional generalized Temperley{Lieb algebras were classied by J. Graham [5] into seven innite families: types A, B, D, E, F , H and I. The rst author was supported in part by an award from the Nueld Foundation. Typeset by A M S-T E X 1 2 R.M. GREEN AND J. LOSONCZY In [7], we showed that a generalized Temperley{Lieb algebra arising from a Coxeter system of arbitrary type admits a canonical (more precisely, an IC) basis. Such a basis is by denition unique; furthermore, it is analogous, in a manner which can be made precise, to G. Lusztig's canonical basis for the ...

Abstract not found

We establish the existence of an IC basis for the generalized Temperley-Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is finite dimensional.

Abstract. We investigate the compatibility of the set of fully commutative elements of a Coxeter group with the various types of Kazhdan–Lusztig cells using a canonical basis for a generalized version of the Temperley–Lieb algebra. Cellules pleinement commutatives de Kazhdan–Lusztig Nous étudions la compatibilité entre l’ensemble des éléments pleinement commutatifs d’un groupe de Coxeter et les divers types de cellules de Kazhdan–Lusztig, en utilisant une base canonique pour une version généralisée de l’algèbre de Temperley– Lieb.