Browsing by Author "Lipp, Johannes"

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    Representations of Hecke algebras of Weyl groups of type A and B
    (2001) Lipp, Johannes; Dipper, Richard (Prof. Dr.)
    The knowledge of the decomposition numbers of Hecke algebras associated to Weyl groups is very useful in the representation theory of finite groups of Lie type since the decomposition matrix of such an algebra embeds into that of the corresponding group. In the investigation of the Hecke algebras themselves, generic constructions - that is, constructions independent of the coefficient ring and the parameters - are a helpful tool. This thesis contributes to those two aspects of the theory of Hecke algebras. The first part of this thesis is concerned with decomposition numbers of blocks of Hecke algebras of type A. In particular, we consider blocks having core (0) and weight 3. First, we derive an upper bound for the decomposition numbers of an arbitrary block. This is used to show that all the decomposition numbers of a block having core (0) and weight 3 are 0 or 1. That result in turn enables us to describe a combinatorial algorithm for their calculation. Furthermore, we show that the decomposition numbers of a block having core (0) and weight 3 depend only on the ordinary and the quantized characteristic of the coefficient field. Moreover, if the ordinary characteristic is neither 2 nor 3 then they are already determined by the quantized characteristic alone. In the second part of this thesis, we construct generic Specht series for Hecke algebras of type A and generic bi-Specht series for Hecke algebras of type B. These are series of right ideals in those algebras such that all subquotients are Specht modules respectively bi-Specht modules. The construction of the Specht series generalizes ideas from Dipper and James for symmetric groups and Hecke algebras of type A. In particular, generic bases for the so-called PK-modules are introduced. The derivation of the bi-Specht series makes use of the Specht series and general methods from Dipper and James for the investigation of Hecke algebras of type B.