On the Irreducible Representations of the Specializations in Characteristics 2 and 3 of the Generic Hecke Algebra of Type F4

Abstract

A complete determination of the irreducible modules of specialized Hecke algebras of type F4, 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=u2 and v=u4 has been obtained by K. Bremke (1994, Manuscripta Math.83, 331–346). In another 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.