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Models and involution models for wreath products and certain Weyl groups. (English) Zbl 0757.20003

A model for the irreducible complex representations of a group \(G\) is a set \(M=\{\rho_ 1,\dots,\rho_ m\}\), where \(\rho_ i: G_ i\to\mathbb{C}\) are linear representations of subgroups of \(G\) such that \(\sum^ m_{i=1}\rho^ G_ i=\sum_{\rho\in \text{Irr.}G}\rho\). In particular we say that \(M\) is an involution model on \(E=\{e_ 1,\dots,e_ m\}\subseteq G\) if the following hold: \(G_ i=C_ G(e_ i)\), \(i=1,\dots,m\), \(\{g\in G: g^ 2=1\}=\prod^ m_{i=1}\{e_ i^ g: g\in G\}\). Similar models are known to exist for the Weyl groups of type \(B_ n\) for all \(n\), and type \(D_ n\) 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 \(H\text{ wr }S_ n\). In this paper the author shows that the Weyl group of type \(D_ 4\) \(((S_ 2\text{ wr }S_ n)\cap A_ 8)\) 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 \(D_ n\).

MSC:

20E22 Extensions, wreath products, and other compositions of groups
20C33 Representations of finite groups of Lie type
20C15 Ordinary representations and characters
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