[tex]D(g)[/tex] is a representaiton of a group denoted by [tex]g[/tex]. The representaion is recucible if it has an invariant subspace, which means that the action of any [tex]D(g)[/tex] on any vector in the subspace is still in the subspace. In terms of a projection operator [tex]P[/tex] onto the subspace this condition can be written as(adsbygoogle = window.adsbygoogle || []).push({});

[tex]PD(g)P=D(g)P~~~~~\forall~g\inG[/tex].

And furthermore the conditon can be converted into

[tex]D(g)P=P~~~~~\forall~g[/tex].

I don't know why the complex condition can be converted into the short and simple one. Can you tell me, thanks a lot.

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# Questions about a projection operator in the representation theoy of groups

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