On the Irreducible Representations of the Specializations in Characteristics 2 and 3 of the Generic Hecke Algebra of type F 4 *

Abstract A complete determination of the irreducible modules of specialized Hecke algebras of type F 4 , with respect to specializations with equal parameters, has been obtained by M. Geck and K. Lux (1991, Manuscripta Math. 70, 285-306) for all characteristics. A similar determination for specializations with v = u 2 and v = u 4 has been obtained by K. Bremke (1994, Manuscripta Math. 83, 331-346). In an earlier paper (1999, J.Algebra 218, 654-671), the authors determined the irreducible modules for all remaining specializations other than those into fields of characteristic 2 or 3, obtaining en route decompositions of the generic irreducible modules under such specializations. In this paper, the corresponding results for characteristic 2 or 3 are obtained. Again, it is found that the decomposition matrices may be expressed in lower uni-triangular form in all these cases and that the splitting fields are those generated by the images of the parameters. * The authors thank the University of Cyprus and the University of Wales, Aberystwyth for supporting this research through exchange visits between th

Hecke algebras of finite type are cellular

Let \cH be the one-parameter Hecke algebra associated to a finite Weyl group $W$, defined over a ground ring in which ``bad'' primes for $W$ are invertible. Using deep properties of the Kazhdan--Lusztig basis of \cH and Lusztig's \ba-function, we show that \cH has a natural cellular structure in the sense of Graham and Lehrer. Thus, we obtain a general theory of ``Specht modules'' for Hecke algebras of finite type. Previously, a general cellular structure was only known to exist in types $A_n$ and $B_n$.Comment: 14 pages; added reference

Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion--convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.Comment: 25 page

Pallikaros, On relations between the classical and the Kazhdan– Lusztig representations of symmetric groups and associated Hecke algebras, J. Pure and Applied Algebra 203

Abstract Let H be the Hecke algebra of a Coxeter system (W, S), where W is a Weyl group of type A n , over the ring of scalars A = Z[q 1/2 , q −1/2 ], where q is an indeterminate. We show that the Specht module S λ , as defined by Dipper and James [6], is naturally isomorphic over A to the cell module of Kazhdan and Lusztig [14] associated with the cell containing the longest element of a parabolic subgroup W J for appropriate J ⊆ S. We give the association between J and λ explicitly. We introduce notions of the T -basis and C-basis of the Specht module and show that these bases are related by an invertible triangular matrix over A. We point out the connection with the work of Garsia and McLarnan [9] concerning the corresponding representations of the symmetric group