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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.

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