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Weights of multipartitions and representations of Ariki-Koike algebras. (English) Zbl 1111.20009

Let 𝔖 n denote the symmetric group on n letters and H n =H n,q (𝔖 n ) be the Iwahori-Hecke algebra corresponding to 𝔖 n . Let G be the complex reflection group C r 𝔖 n . Let 𝔽 be a field. Suppose that q,Q 1 ,,Q r are elements of 𝔽, with q non-zero. The Ariki-Koike algebra n is defined to be the unital associative 𝔽-algebra with presentation

(T i +q)(T i -1)=0(1in-1),(T 0 -Q 1 )(T 0 -Q r )=0,T i T j =T j T i (0i,jn-1,|i-j|>1),T i T i+1 T i =T i+1 T i T i+1 (1in-2);T 0 T 1 T 0 T 1 =T 1 T 0 T 1 T 0 ·

Ariki gave a necessary and sufficient criterion in terms of the parameters q,Q 1 ,,Q r for n to be semi-simple, and described the simple modules in this case. These are indexed by multipartitions of n with r components.

The purpose of this paper is to provide further generalization of the combinatorics of H n to that of n by introducing a notion of ‘weight’ for multipartitions. For each multipartition λ the author defines a non-negative integer called the weight of λ. The author proves some basic properties of this weight function, and examines blocks of small weight.


MSC:
20C08Hecke algebras and their representations
20C30Representations of finite symmetric groups
05E10Combinatorial aspects of representation theory