有关乘积群线性表示的若干结果及其应用

Some Results about the Linear Representadtions of Product Groups and Their Applicadtions

在文献[1]中,J-P Serre给出:当G_(1),G_(2)为有限群时,对任意G_(1)×G_(2)的不可约表示ρ,存在G_(1)、G_(2)的不可约表示分别为ρ_(1)、ρ_(2),使得ρ≌ρ_(1)■ρ_(2).文中主要是对这个结论进行推广:当G_(1),G_(2)为可数无限群,对任意G_(1)×G_(2)的无限维不可约表示ρ,再加上一个自然的条件,就会存在G_(1)、G_(2)的不可约表示分别为ρ_(1)、ρ_(2),使得ρ≌ρ_(1)■ρ_(2).

In the literature[1],J-P Serre gives the result:When G_(1)、G_(2) are finite groups,for any finite dimensional irreducible representadtionρof G_(1)×G_(2),there are irreducible representadtions of ρ_(1) and ρ_(2) as G_(1) and G_(2) respectively,so that ρ≌ρ_(1)■ρ_(2).This article mainly promotes this conclusion:When G_(1)、G_(2) are countable infinite groups,for any infinite dimensional irreducible representadtionρof G_(1)×G_(2),and plus a natural condidtion,then there are irreducible representadtions of ρ_(1) and ρ_(2) as G_(1) and G_(2) respectively,so that ρ≌ρ_(1)■ρ_(2).