## The cyclotomic Jantzen-Schaper theorem

### Gordon James and Andrew Mathas

#### Keywords

Ariki-Koike algebras, cyclotomic
*q*-Schur algebras, Jantzen filtrations.

#### Status

*Trans. AMS*, **352** (2000), 5381-5404.

MR2001b:16017.
#### Abstract

In this paper we use the cyclotomic *q*-Schur algebras to
prove an analogue of the Jantzen-Schaper theorem for the Ariki-Koike
algebras. Most of the argument is devoted to first proving an
analogue of Jantzen's sum formula for the Weyl modules of the
cyclotomic *q*-Schur algebra. The result for the Ariki-Koike
algebras is then deduced by a Schur functor argument. As a
corollary of these results we obtain criteria for the Weyl modules
and Specht modules of these algebras to be irreducible.
As a special case of our results we obtain, for the first time, an
analogue of the Jantzen-Schaper theorem for Coxeter groups of type
**B**.

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*Andrew Mathas*

10th January 1998.