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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={ρ 1 ,,ρ m }, where ρ i :G i are linear representations of subgroups of G such that i=1 m ρ i G = ρIrr.G ρ. In particular we say that M is an involution model on E={e 1 ,,e m }G if the following hold: G i =C G (e i ), i=1,,m, {gG:g 2 =1}= i=1 m {e i g :gG}. 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 HwrS n . In this paper the author shows that the Weyl group of type D 4 ((S 2 wrS n )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:
20E22Extensions, wreath products, and other compositions of groups
20C33Representations of finite groups of Lie type
20C15Ordinary representations and characters of groups