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Zbl 0757.20003
Models and involution models for wreath products and certain Weyl groups.
(English)
[J] J. Lond. Math. Soc., II. Ser. 44, No.1, 55-74 (1991). ISSN 0024-6107; ISSN 1469-7750/e

A model for the irreducible complex representations of a group $G$ is a set $M=\{\rho\sb 1,\dots,\rho\sb m\}$, where $\rho\sb i: G\sb i\to\bbfC$ are linear representations of subgroups of $G$ such that $\sum\sp m\sb{i=1}\rho\sp G\sb i=\sum\sb{\rho\in \text{Irr.}G}\rho$. In particular we say that $M$ is an involution model on $E=\{e\sb 1,\dots,e\sb m\}\subseteq G$ if the following hold: $G\sb i=C\sb G(e\sb i)$, $i=1,\dots,m$, $\{g\in G: g\sp 2=1\}=\prod\sp m\sb{i=1}\{e\sb i\sp g: g\in G\}$. Similar models are known to exist for the Weyl groups of type $B\sb n$ for all $n$, and type $D\sb n$ for $n$ odd. The main result of this paper is to obtain the following generalization to wreath products. Theorem: If a group $H$ has an involution model so does $H\text{ wr }S\sb n$. In this paper the author shows that the Weyl group of type $D\sb 4$ $((S\sb 2\text{ wr }S\sb n)\cap A\sb 8)$ does not have an involution model and he gives a general result on models which gives a model for any of the Weyl groups of type $D\sb n$.
[P.Lakatos (Debrecen)]
MSC 2000:
*20E22 Extensions and other compositions of groups
20C33 Representations of finite groups of Lie type
20C15 Ordinary representations and characters of groups

Keywords: irreducible complex representations; linear representations; Weyl groups; wreath products; involution model

Cited in: Zbl pre06127521

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