A model for the irreducible complex representations of a group

$G$ is a set

$M=\{{\rho}_{1},\cdots ,{\rho}_{m}\}$, where

${\rho}_{i}:{G}_{i}\to \u2102$ 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

$E=\{{e}_{1},\cdots ,{e}_{m}\}\subseteq G$ if the following hold:

${G}_{i}={C}_{G}\left({e}_{i}\right)$,

$i=1,\cdots ,m$,

$\{g\in G:{g}^{2}=1\}={\prod}_{i=1}^{m}\{{e}_{i}^{g}:g\in G\}$. 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

$H\phantom{\rule{4.pt}{0ex}}\text{wr}\phantom{\rule{4.pt}{0ex}}{S}_{n}$. In this paper the author shows that the Weyl group of type

${D}_{4}$ $(\left({S}_{2}\phantom{\rule{4.pt}{0ex}}\text{wr}\phantom{\rule{4.pt}{0ex}}{S}_{n}\right)\cap {A}_{8})$ 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

${D}_{n}$.