Schur elements for the Ariki-Koike algebra and applications

We study the Schur elements associated to the simple modules of the Ariki-Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of type $G(l,p,n)$ in characteristic 0.Comment: The paper contains the results of arXiv:1101.146

Comparison Of a-Functions

. We show agreement of Lusztig's a-function with the a-function in [3]. 1. Introduction In this note, W denotes a finite, simply-laced Weyl group, or the affine Weyl group E 9 = E 8 . Let \Gamma g be the set of simple generators, which are parametrized by the nodes of the associated Coxeter graph \Gamma. Every w 2 W may be written as a product s 1 s 2 s 3 \Delta \Delta \Delta s n of simple generators. If n is minimal, we call this product "reduced" and define l(w) = n and supp(w) to be the set of generators which appear in the product. Let H be the Iwahori-Hecke algebra associated to W . This is an algebra over Q [q 1=2 ; q \Gamma1=2 ] with generators T s for each s 2 \Gamma g satisfying the relations T 2 s = (q \Gamma 1)T s + q, T s T t = T t T s if st = ts, and T s T t T s = T t T s T t if sts = tst, for distinct s, t 2 \Gamma g . Also, let v = q 1=2 . This algebra has a basis Tw , w 2 W , where we have Tw = T s1 \Delta \Delta \Delta T sn whenever s 1 \Delta \Delta \De..

On Kazhdan-Lusztig polynomials for affine Weyl groups and unequal parameters

Let H be the generic Hecke algebra corresponding to an affine Weyl group and unequal parameters. We show the existence of a canonical basis for a certain H-module M"0. The coefficients of the basis elements are generically inverses of the Kazhdan-Lusztig polynomials. We establish a formula for Kazhdan-Lusztig polynomials in terms of certain alcove polynomials. We also express certain Kazhdan-Lusztig polynomials in terms of an analogue of Kostant's partition function and prove an analogue of Kostant's weight multiplicity formula. (orig.)Available from TIB Hannover: RR 1606(96-43) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Reduced words and a length function for G(e,1,n)

In the first part of this paper we study normal forms of elements of the imprimitive complex reflection group G(e,1,n). This allows to prove a conjecture of Broue on basis elements and the canonical symmetrizing form of the associated cyclotomic (Hecke) algebra. Secondly we introduce a root system for G(e,1,n) and study the associated length function. This has many properties in common with the usual length function for finite Weyl groups. (orig.)Available from TIB Hannover: RR 1606(96-24) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Root systems and length functions

We introduce root systems for those imprimitive complex reflection groups which are generated by involutory reflections, and study the associated length functions. These have many properties in common with the usual length functions for finite Weyl groups. (orig.)Available from TIB Hannover: RR 1606(96-55) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman