This paper has two main objectives. First, it has been clear from the beginning of Proctor's work in [P1] that -minuscule

Abstract. We introduce a notion of “freely braided element ” for simply laced Coxeter groups. We show that an arbitrary group element w has at most 2 N(w) commutation classes of reduced expressions, where N(w) is a certain statistic defined in terms of the positive roots made negative by w. This bound is achieved if w is freely braided. In the type A setting, we show that the bound is achieved only for freely braided w. 1.

. 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...

: A generalization of the notion of standard Young tableau has recently arisen from work on the representation theory of ane Hecke algebras. In the generalized setting, a standard tableau is dened to be any element of a nite Weyl group whose inversion set satises a certain pair of intersection conditions. In this paper, we prove that the set of generalized standard tableaux of xed shape, when nonempty, is a certain interval in the weak ordering. In addition, we establish a nonemptiness criterion for the set of standard tableaux of prescribed shape. These results are obtained for shapes that satisfy an integrality condition. Key Words: tableau, Weyl group, ane Hecke algebra. 2 1. Introduction and Definitions Let be a nite crystallographic root system, spanning a real Euclidean space V , and let W be the corresponding Weyl group. The basic facts concerning reection groups and root systems that are used in this paper can be found in [3, 4]. Choose a system + of positive ro...

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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.

Abstract. We continue the study of freely braided elements of simply laced Coxeter groups, which we introduced in a previous work. A known upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is shown to be achieved only when the element is freely braided; this establishes the converse direction of a previous result. It is also shown that a simply laced Coxeter group has finitely many freely braided elements if and only if it has finitely many fully commutative elements. To appear in Advances in Applied Mathematics