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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 $M=\left\{{\rho }_{1},\cdots ,{\rho }_{m}\right\}$, where ${\rho }_{i}:{G}_{i}\to ℂ$ are linear representations of subgroups of $G$ such that ${\sum }_{i=1}^{m}{\rho }_{i}^{G}={\sum }_{\rho \in \text{Irr.}G}\rho$. In particular we say that $M$ is an involution model on $E=\left\{{e}_{1},\cdots ,{e}_{m}\right\}\subseteq G$ if the following hold: ${G}_{i}={C}_{G}\left({e}_{i}\right)$, $i=1,\cdots ,m$, $\left\{g\in G:{g}^{2}=1\right\}={\prod }_{i=1}^{m}\left\{{e}_{i}^{g}:g\in G\right\}$. 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 $H\phantom{\rule{4.pt}{0ex}}\text{wr}\phantom{\rule{4.pt}{0ex}}{S}_{n}$. In this paper the author shows that the Weyl group of type ${D}_{4}$ $\left(\left({S}_{2}\phantom{\rule{4.pt}{0ex}}\text{wr}\phantom{\rule{4.pt}{0ex}}{S}_{n}\right)\cap {A}_{8}\right)$ 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 of groups