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Weights of multipartitions and representations of Ariki-Koike algebras. (English) Zbl 1111.20009
Let $\germ S_n$ denote the symmetric group on $n$ letters and $H_n=H_{n,q}(\germ S_n)$ be the Iwahori-Hecke algebra corresponding to $\germ S_n$. Let $G$ be the complex reflection group $C_r\wr\frak S_n$. Let $\bbfF$ be a field. Suppose that $q,Q_1,\dots,Q_r$ are elements of $\bbfF$, with $q$ non-zero. The Ariki-Koike algebra $\Cal H_n$ is defined to be the unital associative $\bbfF$-algebra with presentation $$\alignat2 (T_i+q)(T_i-1)&=0&\quad &(1\leq i\leq n-1),\\ (T_0-Q_1)\cdots(T_0-Q_r)&=0,&&\\ T_iT_j&=T_jT_i&\quad &(0\leq i,j\leq n-1,\ |i-j|>1),\\ T_iT_{i+1}T_i&=T_{i+1}T_iT_{i+1}&\quad &(1\leq i\leq n-2);\\ T_0T_1T_0T_1&=T_1T_0T_1T_0.&&\endalignat$$ Ariki gave a necessary and sufficient criterion in terms of the parameters $q,Q_1,\dots,Q_r$ for $\Cal H_n$ 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 $\Cal H_n$ by introducing a notion of `weight’ for multipartitions. For each multipartition $\lambda$ the author defines a non-negative integer called the weight of $\lambda$. 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
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Full Text: DOI
References:
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