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Decomposition numbers for Hecke algebras of type : the -separated case
The paper studies the modular representation theory of the cyclotomic Hecke
algebras of type with (\eps,q)-separated parameters. We show that
the decomposition numbers of these algebras are completely determined by the
decomposition matrices of related cyclotomic Hecke algebras of type ,
where and . Furthermore, the proof gives an explicit
algorithm for computing these decomposition numbers. Consequently, in
principle, the decomposition matrices of these algebras are now known in
characteristic zero. In proving these results, we develop a Specht module
theory for these algebras, explicitly construct their simple modules and
introduce and study analogues of the cyclotomic Schur algebras of type
when the parameters are (\eps,q)-separated. The main results of
the paper rest upon two Morita equivalences: the first reduces the calculation
of all decomposition numbers to the case of the \textit{-splittable
decomposition numbers} and the second Morita equivalence allows us to compute
these decomposition numbers using an analogue of the cyclotomic Schur algebras
for the Hecke algebras of type .Comment: Final versio
Wild division algebras over Laurent series fields
In this paper we study some special classes of division algebras over a
Laurent series field with arbitrary residue field. We call the algebras from
these classes as splittable and good splittable division algebras. It is shown
that these classes contain the group of tame division algebras. For the class
of good division algebras a decomposition theorem is given. This theorem is a
generalization of the decomposition theorems for tame division algebras given
by Jacob and Wadsworth. For both clases we introduce a notion of a -map
and develop a technique of -maps for division algebras from these
classes. Using this technique we reprove several old well known results of
Saltman and get the positive answer on the period-index conjecture of M.Artin:
the exponent of is equal to its index for any division algebra over a
-field , when F\eq F_1((t_2)), where is a -field. The
paper includes also some other results about splittable division algebras,
which, we hope, will be useful for the further investigation of wild division
algebras.Comment: 32 page
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